# Gelfand-Fuchs cohomology for affine superspaces $\mathbb{A}^{n,1}$

Slava Pimenov

###### Contents

  1. 0 Introduction
  2. 1 Recollections
  3. 2 Cohomology of $\mathfrak{gl}(n,1)$
  4. 3 Cohomology of $\mathcal{V}_{n,1}$

##  0 Introduction

Let $M$ be a differentiable (or algebraic) manifold, and denote by
$\mathcal{V}_{M}=\Gamma(M,TM)$ the space of global section of the tangent
bundle. The Lie bracket of vector fields on $M$ equips $\mathcal{V}_{M}$ with
the structure of Lie algebra, and we are interested in the cohomology
$H^{\bullet}(\mathcal{V}_{M},\mathbf{k})$. As is often the case with
cohomology theories the calculation of
$H^{\bullet}(\mathcal{V}_{M},\mathbf{k})$ can be split into two steps.

  1. a)

Local calculation of
$H^{\bullet}(\mathcal{V}_{\widehat{\mathbb{A}}^{n}},\mathbf{k})$, where
$\widehat{\mathbb{A}}^{n}$ is a formal $n$-dimensional disk.

  2. b)

Some kind of local to global construction.

The local calculation is a classical result and can be found in [Fuk].
Consider the topological group $GL(n,\mathbb{C})$, and let $BGL(n)$ be its
classifying space. Denote by $p\colon EGL(n)\to BGL(n)$ the tautological
principal $GL(n)$-bundle over the classifying space. We will also write
$\mathop{\mathrm{sk}}_{d}BGL(n)$ for the $d$-dimensional skeleton of $BGL(n)$,
i.e. the subspace formed by all cells of dimension up to $d$. Let
$X_{d}=p^{-1}(\mathop{\mathrm{sk}}_{d}BGL(n))$ be the fiber of the
tautological bundle over the $d$-dimensional skeleton.

[Gelfand-Fuchs] We have an isomorphism

$H^{\bullet}(\mathcal{V}_{\widehat{\mathbb{A}}^{n}},\mathbf{k})\ \simeq\
H^{\bullet}(X_{2n},\mathbf{k}).$

The local to global construction can be produced by considering the spectral
sequence associated to the diagonal filtration on the cochain complex
$C^{\bullet}(\mathcal{V}_{M},\mathbf{k})$, which is formed by cochains
supported on various diagonal subspaces of $M^{q}$ and filtered by inclusion
of diagonals ([GF]). A more modern approach to this construction uses the
formalism of factorization algebras ([HK]).

[Supermanifolds.] Now let $M$ be a (smooth or algebraic) supermanifold, in
this case even the local picture of the completion of the affine superspace
$\widehat{\mathbb{A}}^{m,n}$ is not fully understood. For convenience we will
write $\mathcal{V}_{m,n}=\mathcal{V}_{\widehat{\mathbb{A}}^{m,n}}$. Some
partial results have been obtained regarding the cohomology
$H^{\bullet}(\mathcal{V}_{m,n},\mathbf{k})$ ([AF]) as well as some results
concerning cohomology with other coefficients and cohomology of contact vector
fields ([BAF], [FK]).

Let us recall what is known about cohomology of $\mathcal{V}_{m,n}$. The first
part of the next theorem was announced without proof in [FL] and independently
proved by Koszul in [Ko]. In its entirety it was shown in [AF].

[Astashkevich-Fuchs] For Lie superalgebras $\mathcal{V}_{m,n}$ with $m<n$ we
have

  1. a)

$H^{\bullet}(\mathcal{V}_{0,n},\mathbf{k})\simeq
H^{\bullet}(S^{2n-1},\mathbf{k})$,

  2. b)

the inclusion $\mathcal{V}_{0,n}\hookrightarrow\mathcal{V}_{m,n}$ induces an
isomorphism $H^{\bullet}(\mathcal{V}_{m,n},\mathbf{k})\simeq
H^{\bullet}(\mathcal{V}_{0,n},\mathbf{k})$.

Here $S^{2n-1}$ denotes the $(2n-1)$-dimensional sphere. The first part is
established using a similar argument to that of the classical case of
$\mathcal{V}_{m,0}$, and the second part is established by comparing cochain
complexes for $\mathcal{V}_{0,n}$ and $\mathcal{V}_{m,n}$. Furthermore, this
method can be pushed one step further to obtain the non-trivial result.

[Astashkevich-Fuchs] We have an isomorphism

$H^{\bullet}(\mathcal{V}_{n,n},\mathbf{k})\ \simeq\
H^{\bullet}(\SS^{2n}GL(n,\mathbb{C}),\mathbf{k}).$

Here $\SS$ denotes the topological suspension functor. At this point it is
unclear if this method can be used to continue further and include cases of
$\mathcal{V}_{m,n}$ with $m>n$. Using a different approach, more in line with
the standard calculation used in the classical case, we calculate the
cohomology for $\mathcal{V}_{n,1}$. The main result of this paper is the
following theorem.

We have an isomorphism

$H^{\bullet}(\mathcal{V}_{n,1},\mathbf{k})\ \simeq\
H^{\bullet}(\SS^{2}X_{2(n-1)},\mathbf{k}).$

Comparing these results we are led to the following conjectural form for the
general case. This conjecture is well known among the specialists but doesn’t
seem to appear in the literature.

######  Conjecture .

For Lie superalgebras $\mathcal{V}_{m,n}$ with $m,n\geqslant 0$ we have

$H^{\bullet}(\mathcal{V}_{m,n},\mathbf{k})\ \simeq\
H^{\bullet}(\SS^{2n}X_{2(m-n)},\mathbf{k}).$

Here as before $X_{2(m-n)}$ is the restriction of the tautological principal
bundle over $BGL(m)$ to the $2(m-n)$-dimensional skeleton. Clearly, for $n=0$
and $n=1$ the conjecture matches the theorems  ‣ 0 Introduction,  ‣ 0
Introduction. For $m=n$ we observe that the $0$-skeleton
$\mathop{\mathrm{sk}}_{0}BGL(n)$ is a point and the fiber of the tautological
fibration $p$ is $GL(n)$, hence we recover statement of theorem  ‣ 0
Introduction. In the case $m<n$ the skeleton
$\mathop{\mathrm{sk}}_{2(m-n)}BGL(m)$ is an empty set, and we put by
definition the suspension of an empty set to be the zero-dimensional sphere
$S^{0}$. Therefore, $\SS^{2n}X_{2(m-n)}=S^{2n-1}$ and we arrive to the
statement of theorem  ‣ 0 Introduction.

[Outline of the proof.] Similar to the classical case we begin the proof by
considering Lie subalgebra
$\mathfrak{gl}(m,n)\hookrightarrow\mathcal{V}_{m,n}$ and constructing the
associated spectral sequence starting from cohomology of $\mathfrak{gl}(m,n)$
and converging to the cohomology of $\mathcal{V}_{m,n}$. This is done in the
beginning of section 3. We arrive to the question of describing the space of
invariants in the tensor product

$\Lambda^{p}(V)\mathop{\otimes}\limits\Lambda^{p}(S^{2}(V^{*})\mathop{\otimes}\limits
V).$

In the classical case Gelfand and Fuchs introduce multiplicative generators of
this space of invariants using graph description. Namely, they define
invariants $\psi_{r}$ represented by the graphs of the following type.

$\scriptstyle 1$$\scriptstyle 2$$\scriptstyle
3$$\scriptstyle\cdots$$\scriptstyle r$

Here vertices with only one output correspond to factors $V$ and vertices with
two inputs and one output to factors $S^{2}(V^{*})\mathop{\otimes}\limits V$.
Of course, the same $\psi_{r}$ could be taken as generators of invariants in
the case of $\mathcal{V}_{m,n}$ as well, however this description is ill
suited to describe the relations between them. Instead, we adopt a different
approach using Schur functors $\Sigma^{\lambda}$. Each invariant comes from
the one-dimensional invariant subspace
$\mathbf{k}\hookrightarrow\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$,
for some diagram $\lambda$ and we identify all diagrams that contribute to the
space of invariants.

The next difficulty to resolve is the fact that unlike the classical case, the
category of representations of $\mathfrak{gl}(m,n)$ in weight $0$ is not
semisimple. Therefore, it is not enough just to know the space of invariants
in order to calculate the first layer of the spectral sequence, we need to
know exactly what diagram $\lambda$ the invariant came from. Here we restrict
our attention to the case $\mathfrak{gl}(n,1)$ and perform the necessary
calculations in section 2. This is a major difficulty of the super-case
compared to the classical one. In order to apply the methodology of this paper
to $\mathcal{V}_{m,n}$ for general $m\geqslant n$ we need to have an analog of
lemma 2.1 for $\mathfrak{gl}(n,n)$.

Finally, equipped with these two results we complete the proof by analyzing
the spectral sequence associated to the Lie subalgebra
$\mathfrak{gl}(n,1)\hookrightarrow\mathcal{V}_{n,1}$.

The author would like to thank Dmitry Fuchs, Mikhail Kapranov and Vera
Serganova who looked at the preliminary version of the paper and provided
valuable remarks.

##  1 Recollections

In this section we introduce notations and recall some basic facts that will
be used in this paper. We denote by $\mathbf{k}$ an algebraically closed field
of characteristic $0$.

[Schur functors.] Let $S_{n}$ be a symmetric group on $n$ elements, and
$\mathbf{k}[S_{n}]$ its group algebra. The irreducible representations of
$S_{n}$ are parametrized by the conjugacy classes of $S_{n}$, and each such
class corresponds to an unordered partition
$\lambda=(\lambda_{1},\ldots,\lambda_{k})$ of $n$, with
$\lambda_{1}\geqslant\lambda_{2}\geqslant\ldots\geqslant\lambda_{k}>0$ and
$\sum_{i=1}^{k}\lambda_{i}=n$. If we write a representative of the conjugacy
class as a product of cycles, then the integers $\lambda_{i}$ are the lengths
of the cycles. We will denote the corresponding irreducible $S_{n}$-module by
$M_{\lambda}$.

It is often convenient to write such a partition $\lambda$ as a Young diagram
consisting of $k$ rows such that row $i$ contains $\lambda_{i}$ cells. We will
call the number $k$ the height of the diagram $\lambda$ and denote it by
$\mathrm{ht}(\lambda)$. Furthermore, we will write $\lambda^{\prime}$ for the
transposed Young diagram, i.e. a diagram obtained from $\lambda$ by reflecting
it along the diagonal. More formally, we put
$\lambda^{\prime}_{j}=\max\\{i\mid\lambda_{i}\geqslant j\\}$.

Consider a symmetric monoidal category
$(\mathcal{C},\mathop{\otimes}\limits)$, i.e. a monoidal category such that
for any two objects $V,W\in\mathcal{C}$ we have natural isomorphisms
$c_{V,W}\colon V\mathop{\otimes}\limits W\to W\mathop{\otimes}\limits V$
compatible with the monoidal structure and such that the composition
$c_{W,V}\circ c_{V,W}=\mathrm{id}_{V\mathop{\otimes}\limits W}$. Using
isomorphisms $c_{V,V}$ the $n$-fold tensor product $V^{\mathop{\otimes}\limits
n}$ is equipped with the action of $S_{n}$ and for each partition $\lambda$
the Schur functor $\Sigma^{\lambda}$ is defined by

$\Sigma^{\lambda}(V)=V^{\mathop{\otimes}\limits
n}\mathop{\otimes}\limits_{\mathbf{k}[S_{n}]}M_{\lambda}.$

We also denote by $S^{n}$ and $\Lambda^{n}$ the symmetric and exterior powers
respectively and recall that we have isomorphisms of
$(\mathfrak{gl}(V)\times\mathfrak{gl}(W))$-modules

$S^{n}(V\mathop{\otimes}\limits
W)=\bigoplus_{|\lambda|=n}\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(W),\quad\quad\Lambda^{n}(V\mathop{\otimes}\limits
W)=\bigoplus_{|\lambda|=n}\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda^{\prime}}(W).$

[Super vector spaces.] A super vector space over $\mathbf{k}$ is a
$\mathbb{Z}/2$-graded vector space $V=(V_{0},V_{1})$. We denote by
$\mathrm{sVect}$ the symmetric monoidal category of such spaces with the
tensor product given by

$V\mathop{\otimes}\limits W=(V_{0}\mathop{\otimes}\limits W_{0}\oplus
V_{1}\mathop{\otimes}\limits W_{1},V_{1}\mathop{\otimes}\limits W_{0}\oplus
V_{0}\mathop{\otimes}\limits W_{1}),$

and the symmetry map given by the Koszul sign rule, i.e. for any two
homogeneous elements $v\in V$ and $w\in W$ of degrees $\bar{v}$, $\bar{w}$
respectively we have

$c_{V,W}(v\mathop{\otimes}\limits
w)=(-1)^{\bar{v}\bar{w}}w\mathop{\otimes}\limits v.$

The dimension of super vector space $V$ is a pair of integers $(m,n)$, where
$m=\mathop{\mathrm{dim}}\nolimits V_{0}$ and $n=\mathop{\mathrm{dim}}\nolimits
V_{1}$. The Schur functor $\Sigma^{\lambda}(V)$ is non-zero if and only if the
diagram $\lambda$ is contained in a thick hook with $m$ rows and $n$ columns,
in other words $\lambda_{i}\leqslant n$ whenever $i>m$.

[Lie superalgebra $\mathfrak{gl}(m,n)$.] A Lie superalgebra $\mathfrak{g}$ is
a Lie algebra in the symmetric monoidal category $\mathrm{sVect}$. Explicitly,
it is a super vector space equipped with the bracket satisfying the
commutation relation $[x,y]=-(-1)^{\bar{x}\bar{y}}[y,x]$ and the Jacobi
relation

$[[x,y],z]=[x,[y,z]]-(-1)^{\bar{x}\bar{y}}[y,[x,z]],$

for any three homogeneous elements $x,y,z\in\mathfrak{g}$.

Let $V$ be a super vector space of dimension $(m,n)$, we denote by
$\mathfrak{gl}(m,n)$ the Lie superalgebra of all $\mathbf{k}$-endomorphisms of
$V$ (that do not necessarily preserve the $\mathbb{Z}/2$ grading). Let us fix
a basis $\\{v_{i}\\}$ of $V$ and write $e_{ij}=v_{i}\mathop{\otimes}\limits
v_{j}^{*}$ for the corresponding basis of $\mathfrak{gl}(m,n)$. Clearly, we
have $\bar{e}_{ij}=\bar{v}_{i}+\bar{v}_{j}\in\mathbb{Z}/2$.

This Lie superalgebra is equipped with the non-degenerate invariant super-
symmetric bilinear form, defined by

$(e_{ij},e_{ji})=(-1)^{\bar{v}_{i}},$

and with all other pairings equal to $0$. Denote by
$\mathfrak{h}\subset\mathfrak{gl}(m,n)$ the abelian subalgebra of diagonal
matrices, the restriction of this bilinear form to $\mathfrak{h}$ is also non-
degenerate, and we use it to identify $\mathfrak{h}$ with $\mathfrak{h}^{*}$.
Let $\alpha\in\mathfrak{h}$ be a root of $\mathfrak{gl}(m,n)$, we denote
$H_{\alpha}$ the hyperplane in $\mathfrak{h}$ orthogonal to $\alpha$. We say
that the root $\alpha$ is isotropic if $\alpha\in H_{\alpha}$.

Consider the group $S_{m}\times S_{n}$ acting on the space of functions
$\mathbf{k}[\mathfrak{h}]$ by permuting the first $m$ and the last $n$
coordinates. Then we have the following description of the center
$Z\mathfrak{gl}(m,n)$ of the universal enveloping algebra
$U\mathfrak{gl}(m,n)$ using the Harish-Chandra map $\psi$.

######  Theorem 1.1 ([Mu]).

The map $\psi\colon Z\mathfrak{gl}(m,n)\to\mathbf{k}[\mathfrak{h}]$ is a
monomorphism and the image consists of $(S_{m}\times S_{n})$-invariant
functions $f$ such that $f(\lambda)=f(\lambda+t\alpha)$ for any
$t\in\mathbf{k}$, whenever $\lambda\in H_{\alpha}$ for an isotropic root
$\alpha$.

[Cohomology of $\mathfrak{gl}(m,n)$.] Let $\mathfrak{g}=\mathfrak{gl}(m,n)$,
and $V$ as before the standard representation of $\mathfrak{g}$. We will be
interested in cohomology spaces
$H^{\bullet}(\mathfrak{g},\Sigma^{\alpha}V\mathop{\otimes}\limits\Sigma^{\beta}V^{*})$.
First, denote by $h$ the identity element of $\mathfrak{g}$. Since the Lie
algebra acts trivially on its cohomology we immediately see that this
cohomology space can only be nontrivial if $|\alpha|=|\beta|$. Furthermore,
the Hom-space
$\mathbb{R}\mathrm{Hom}_{\mathfrak{g}}(\Sigma^{\beta}V,\Sigma^{\alpha}V)$ can
be non-zero only if the two $\mathfrak{g}$-modules $\Sigma^{\alpha}V$ and
$\Sigma^{\beta}V$ have the same generalized central character (i.e. supported
at the same point in $\mathop{\mathrm{Spec}}\nolimits Z\mathfrak{g}$).
However, due to the description of the center above and the fact that all
$\alpha_{i}$ and $\beta_{i}$ are non-negative this can only happen if
$\alpha=\beta$.

Consider a Lie subalgebra $\mathfrak{g}^{\prime}\subset\mathfrak{g}$, and
denote by
$\mathop{\mathrm{missing}}{Ind}\nolimits_{\mathfrak{g}^{\prime}}^{\mathfrak{g}}\colon\mathfrak{g}^{\prime}\textrm{-}\mathrm{Mod}\to\mathfrak{g}\textrm{-}\mathrm{Mod}$
the induction functor

$\mathop{\mathrm{missing}}{Ind}\nolimits_{\mathfrak{g}^{\prime}}^{\mathfrak{g}}(M)=\mathrm{Hom}_{U\mathfrak{g}^{\prime}}(U\mathfrak{g},M).$

######  Theorem 1.2.

Let $\mathfrak{g}=\mathfrak{gl}(m,n)$ and assume $m\geqslant n$, then

  1. a)

The inclusion $\mathfrak{gl}(m)\hookrightarrow\mathfrak{g}$ induces
isomorphism of cohomology

$H^{\bullet}(\mathfrak{g},\mathbf{k})\simeq
H^{\bullet}(\mathfrak{gl}(m),\mathbf{k})\simeq\Lambda[e_{1},e_{3},\ldots,e_{2m-1}],$

where each generator $e_{i}$ is of cohomological degree $i$.

  2. b)

Let
$\mathfrak{g}_{0}=\mathfrak{gl}(m)\oplus\mathfrak{gl}(n)\hookrightarrow\mathfrak{g}$
be the even Lie subalgebra of $\mathfrak{g}$, then

$H^{\bullet}(\mathfrak{g},\mathop{\mathrm{missing}}{Ind}\nolimits_{\mathfrak{g}_{0}}^{\mathfrak{g}}(\mathbf{k}))\simeq
H^{\bullet}(\mathfrak{g}_{0},\mathbf{k})\simeq\Lambda[e_{1},e_{3},\ldots,e_{2m-1}]\mathop{\otimes}\limits\Lambda[e^{\prime}_{1},e^{\prime}_{3},\ldots,e^{\prime}_{2n-1}].$

Proof: The first part is the well known calculation and we refer to [Fuk] for
details. The second follows immediately from the definition of the induction
functor and the fact that $U\mathfrak{g}$ is a free
$U\mathfrak{g}_{0}$-module. $\square$

[Lie superalgebra $\mathcal{V}_{m,n}$.] Consider affine superspace
$\mathbb{A}^{m,n}=\mathop{\mathrm{Spec}}\nolimits\mathcal{O}_{\mathbb{A}^{m,n}}$,
where

$\mathcal{O}_{\mathbb{A}^{m,n}}=\mathbf{k}[x_{1},\ldots
x_{m},\xi_{1},\ldots\xi_{n}]$

is the polynomial super-algebra of functions with even coordinates $x_{i}$ and
odd coordinates $\xi_{j}$. We denote by
$\widehat{\mathcal{O}}_{\mathbb{A}^{m,n}}$ its completion at point
$0\in\mathbb{A}^{m,n}$, i.e.
$\widehat{\mathcal{O}}_{\mathbb{A}^{m,n}}=\mathbf{k}[[x_{1},\ldots,x_{m},\xi_{1},\ldots,\xi_{n}]]$,
equipped with the power series topology. We will be interested in the Lie
superalgebra of continuous self derivations

$\mathcal{V}_{m,n}=\mathrm{Der}_{\mathrm{cont}}(\widehat{\mathcal{O}}_{\mathbb{A}^{m,n}}).$

Explicitly, the Lie superalgebra $\mathcal{V}_{m,n}$ consists of elements of
the form $\sum f_{i}{\partial_{x_{i}}}+\sum g_{j}{\partial_{\xi_{j}}}$, with
the obvious bracket. It contains the subalgebra $\mathfrak{gl}(m,n)$ spanned
by linear derivations
$\\{x_{i}\partial_{x_{j}},x_{i}\partial_{\xi_{j}},\xi_{i}\partial_{x_{j}},\xi_{i}\partial_{\xi_{j}}\\}$.

[Spectral sequence.] Let $\mathfrak{g}$ be a Lie superalgebra,
$\mathfrak{h}\hookrightarrow\mathfrak{g}$ a Lie subalgebra and $M$ a
$\mathfrak{g}$-modules. The chain complex
$\Lambda^{\bullet}\mathfrak{g}\mathop{\otimes}\limits M$ is equipped with an
increasing filtrations by the number of elements from $\mathfrak{h}$, which
induces a decreasing filtration on the cochain complex
$C^{\bullet}(\mathfrak{g},M)$. We have the spectral sequence associated to
this filtration

$E_{1}^{pq}=H^{q}(\mathfrak{h},\
\mathrm{Hom}(\Lambda^{p}(\mathfrak{g}/\mathfrak{h}),M)\ )\Rightarrow
H^{p+q}(\mathfrak{g},M).$

This will be the main computational tool in this paper, both for the
cohomology of $\mathfrak{gl}(n,1)$ as well as for the cohomology of
$\mathcal{V}_{n,1}$.

##  2 Cohomology of $\mathfrak{gl}(n,1)$

Throughout this section we consider $\mathfrak{g}=\mathfrak{gl}(n,1)$, and $V$
will be the standard representation of $\mathfrak{gl}(n,1)$. The section is
dedicated to calculating the cohomology
$H^{\bullet}(\mathfrak{g},\Sigma^{\alpha}V\mathop{\otimes}\limits\Sigma^{\alpha}V^{*})$.
We will proceed by induction on $n$, with the case $n=1$ being the base of
induction. For $\mathfrak{gl}(1,1)$ the calculation can be done directly,
since we have a complete description of the category of $\mathfrak{g}$-modules
in this case ([FKV]).

[Representations of $\mathfrak{gl}(1,1)$.] We begin be recalling relevant
details concerning the category of $\mathfrak{gl}(1,1)$-modules. As before we
put $h\in\mathfrak{g}=\mathfrak{gl}(1,1)$ to be the identity element, and put

$e=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix},\quad f=\begin{pmatrix}0&0\\\
1&0\end{pmatrix},\quad g=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$

The category of $\mathfrak{g}$-modules in non-zero $h$-weight $\lambda$ is
semisimple, and the simple modules are two-dimensional with basis vectors of
$(h,g)$-weights $(\lambda,\gamma)$ and $(\lambda,\gamma-2)$ and the action of
$e$ and $f$ are as follows.

${\mathop{\bullet}\limits_{\gamma-2}^{\lambda}}$${\mathop{\bullet}\limits_{\gamma}^{\lambda}}$$\scriptstyle{e}$$\scriptstyle{f}$

We write $V_{\lambda,\gamma}^{\overline{0}}$ for the simple module as above if
the vector of $g$-weight $\gamma$ is even and
$V_{\lambda,\gamma}^{\overline{1}}$ if it is odd. In particular we have
$V=V_{1,1}^{\overline{0}}$. Consider a hook diagram $\lambda=(c,1,\ldots,1)$,
of height $r$ (so that $|\lambda|=c+r-1$). We have

$\Sigma^{\lambda}(V)=V_{c+r-1,c-r+1}^{\overline{r-1}}.$

The tensor product
$\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$ is
isomorphic to the induced module
$\mathop{\mathrm{missing}}{Ind}\nolimits_{\mathfrak{g}_{0}}^{\mathfrak{g}}(\mathbf{k})$.

${\bullet}$${\bullet}$${\bullet}$${\bullet}$$\scriptstyle{e}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{-e}$

######  Lemma 2.1.

Let $\mathfrak{g}=\mathfrak{gl}(1,1)$ and $V$ the standard representation of
$\mathfrak{g}$, then

$H^{\bullet}(\mathfrak{g},\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*}))=\begin{cases}\mathbf{k}[e_{1}],&\text{if
$|\lambda|=0$,}\\\
\mathbf{k}[e_{1},e^{\prime}_{1}],&\text{otherwise},\end{cases}$

where both $e_{1}$ and $e^{\prime}_{1}$ are of cohomological degree $1$.

Proof: The statement follows immediately from the discussion above and theorem
1.2.

$\square$

######  Proposition 2.2.

Let $\mathfrak{g}=\mathfrak{gl}(n,1)$, and $V$ the standard representation of
$\mathfrak{g}$, then

$H^{\bullet}(\mathfrak{g},\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*}))=\begin{cases}\mathbf{k}[e_{1},\ldots,e_{2n-1}],&\text{if
$\mathrm{ht}(\lambda)\leqslant n-1$,}\\\
\mathbf{k}[e_{1},\ldots,e_{2n-1},e^{\prime}_{1}],&\text{otherwise},\end{cases}$

where generators $e_{i}$ are of cohomological degree $i$ and $e^{\prime}_{1}$
is of degree $1$.

Proof: The case $n=1$ is the statement of the previous lemma. Assume that
$n>1$ and that the proposition has been established for all
$\mathfrak{gl}(m,1)$ with $m<n$. Consider the Lie subalgebra
$\mathfrak{h}=\mathfrak{gl}(1)\oplus\mathfrak{gl}(n-1,1)\hookrightarrow\mathfrak{g}$
embedded as diagonal blocks. We will write $E$ for the standard representation
of $\mathfrak{gl}(1)$ (it is of course a one-dimensional even vector space,
however we use letter $E$ instead of $\mathbf{k}$ to distinguish it from the
trivial representation), and $W$ for the standard representation of
$\mathfrak{gl}(n-1,1)$. We have

$\mathfrak{g}/\mathfrak{h}\ \simeq\ E\mathop{\otimes}\limits W^{*}\oplus
E^{*}\mathop{\otimes}\limits W,$

and therefore

$\Lambda^{p}(\mathfrak{g}/\mathfrak{h})\ \simeq\
\bigoplus_{i+j=p}S^{i}(E)\mathop{\otimes}\limits\Lambda^{i}(W^{*})\mathop{\otimes}\limits
S^{j}(E^{*})\mathop{\otimes}\limits\Lambda^{j}(W).$

Furthermore, we have $V=E\oplus W$ and hence

(2.2.1) $\Sigma^{\lambda}(V)\ \simeq\
\bigoplus_{\beta}S^{k}(E)\mathop{\otimes}\limits\Sigma^{\beta}(W),$

where the sum is taken over all subdiagrams $\beta\subset\lambda$, such that
the skew diagram $\lambda\backslash\beta$ can be collapsed into a single row
diagram, and $k$ is the size of $\lambda\backslash\beta$.

Consider the spectral sequence for the subalgebra
$\mathfrak{h}\hookrightarrow\mathfrak{g}$:

$E_{1}^{pq}=H^{q}(\mathfrak{h},\mathrm{Hom}(\Lambda^{p}(\mathfrak{g}/\mathfrak{h}),\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})))\Rightarrow
H^{p+q}(\mathfrak{g},\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})).$

Using the decompositions described above we see that the first layer of this
spectral sequence consists of terms of the form

(2.2.2)
$H^{\bullet}(\mathfrak{gl}(1)\oplus\mathfrak{gl}(n-1,1),S^{k}(E)\mathop{\otimes}\limits
S^{l}(E^{*})\mathop{\otimes}\limits\Sigma^{\beta}(W)\mathop{\otimes}\limits\Sigma^{\gamma}(W^{*})).$

From the discussion in paragraph 1.1 it follows that these terms are non-zero
only if $\gamma=\beta$ (in which case we necessarily have $k=l$). We will
refer to the sum of all such terms as the component of type $\beta$. For
convenience we introduce notation
$\Delta^{\lambda}(V)=\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$.

[Case of $\mathrm{ht}(\lambda)\geqslant n$.] For any Young diagram $\alpha$ we
can construct diagram $\alpha^{+}$ by adding one box in the first column.
Recall that for the Schur functor $\Sigma^{\lambda}(V)$ to be non-zero the
diagram $\lambda$ must have no more than $n$ boxes in the second column, and
similarly diagrams $\beta$ in (2.2.1) and (2.2.2) no more than $(n-1)$ boxes.
Using this we observe that the diagram combinatorics in the first layer of
spectral sequences for modules $\Delta^{\lambda}(V)$ and
$\Delta^{\lambda^{+}}(V)$ are identical, in the sense that we have a bijection
between terms of type $\beta$ in the former and terms of type $\beta^{+}$ in
the latter. Moreover, since $\mathrm{ht}(\beta)\geqslant(n-1)$ and by
inductive assumption we have isomorphism of cohomology

$H^{\bullet}(\mathfrak{gl}(n-1,1),\Delta^{\beta}(W))\simeq
H^{\bullet}(\mathfrak{gl}(n-1,1),\Delta^{\beta^{+}}(W))$

we see that the two spectral sequences are isomorphic and converge to the same
result. Therefore it is enough to consider the case $\mathrm{ht}(\lambda)=n$.

We split the first layer $E_{1}$ into two parts, the first one
$E^{\prime}_{1}$ will consist of all components of type $\beta$ with
$\mathrm{ht}(\beta)=n-1$, and the second one $E^{\prime\prime}_{1}$ will
consist of all the remaining components (i.e. components of type $\beta$ with
$\mathrm{ht}(\beta)\geqslant n$).

Consider also Lie subalgebra
$\mathfrak{k}=\mathfrak{gl}(1)\oplus\mathfrak{gl}(n-1)\hookrightarrow\mathfrak{gl}(n)$,
and let $V_{0}$ and $W_{0}$ be the standard representations of
$\mathfrak{gl}(n)$ and $\mathfrak{gl}(n-1)$ respectively. We have the
corresponding spectral sequence

$F_{1}^{pq}=H^{q}(\mathfrak{k},\mathrm{Hom}(\Lambda^{p}(\mathfrak{gl}(n)/\mathfrak{k}),\Delta^{\lambda}(V_{0})))\Rightarrow
H^{p+q}(\mathfrak{gl}(n),\Delta^{\lambda}(V_{0})).$

The latter cohomology is isomorphic to the cohomology of the trivial
$\mathfrak{gl}(n)$-module, and as was mentioned in theorem 1.2 it is
isomorphic to the exterior algebra $\mathbf{k}[e_{1},\ldots,e_{2n-1}]$.

Observe that the diagram combinatorics of the part $E^{\prime}_{1}$ and
$F_{1}$ are identical, and moreover by inductive assumption we have

$H^{\bullet}(\mathfrak{gl}(n-1,1),\Delta^{\beta}(W))\simeq
H^{\bullet}(\mathfrak{gl}(n-1),\Delta^{\beta}(W_{0}))\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{1}].$

So if we show that the second part $E^{\prime\prime}_{1}$ is acyclic we will
conclude that the spectral sequence $E_{\bullet}$ converges to
$\mathbf{k}[e_{1},\ldots,e_{2n-1}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{1}]$
as required.

In fact we will show that every component of type $\beta$ in
$E^{\prime\prime}_{1}$ is acyclic. The process described in the beginning or
the proof by which diagram $\beta$ is obtained from $\lambda$ consists of
choosing a common subdiagram $\alpha$ such that the difference
$\lambda\backslash\alpha$ is collapsible to a single row diagram and the
difference $\beta\backslash\alpha$ collapses to a single column diagram. We
denote $k=|\lambda\backslash\alpha|$. Of course there could be multiple such
ways to obtain $\beta$ from $\lambda$, so let us pick $\alpha$ with minimal
$k$ (such $\alpha$ is unique). We call a box in $\alpha$ flippable if by
removing it we obtain a diagram $\alpha_{1}$ which provides another way to
construct $\beta$ out of $\lambda$. It is easy to see that a box is flippable
if and only if it has no boxes to the right in $\beta$ and has no boxes on top
of it in $\lambda$. We denote by $d=d(\beta,\lambda)$ the number of flippable
boxes. Since $\mathrm{ht}(\lambda)=n$ and $\mathrm{ht}(\beta)\geqslant n$ we
see that $d>0$.

By construction of the spectral sequence the differential in $E_{1}$
corresponds to the process of flipping boxes, therefore we find that the
component of $E^{\prime\prime}_{1}$ of type $\beta$ is of the form

$\Delta^{\beta}(W)\mathop{\otimes}\limits\bigotimes_{1}^{2d}\left(\mathbf{k}\to\mathbf{k}\right).$

The $2d$ factors come from $d$ flippable boxes involving terms
$\Sigma^{\beta}(W)$ and another $d$ for terms $\Sigma^{\beta}(W^{*})$. Since
each factor is acyclic and $d>0$ we conclude that the component of
$E^{\prime\prime}_{1}$ of type $\beta$ is acyclic.

This concludes the proof of the second part of the proposition.

[Case of $\mathrm{ht}(\lambda)\leqslant n-1$.] As before we split the first
layer of the spectral sequence into two parts $E^{\prime}_{1}$ consisting of
components of type $\beta$ with $\mathrm{ht}(\beta)\leqslant n-1$, and
$E^{\prime\prime}_{1}$ consisting of components with
$\mathrm{ht}(\beta)\geqslant n$.

First, we compare $E^{\prime}_{1}$ with $F_{1}$. Using the inductive
assumption we see that components of type $\beta$ with
$\mathrm{ht}(\beta)<n-1$ provide isomorphic terms in $E^{\prime}_{1}$ and
$F_{1}$, while those with $\mathrm{ht}(\beta)=n-1$ have extra summand of the
form $\mathbf{k}[e_{1},\ldots,e_{2n-3}]e^{\prime}_{1}z_{\beta}$, where
$z_{\beta}$ is just a basis vector introduced for bookkeeping purposes.
Moreover, from the discussion at the end of the last paragraph we see that
these extra terms survive into $E_{2}$ if and only if $\beta$ has no flippable
boxes, i.e. $d(\beta,\lambda)=0$. We will show that $E^{\prime\prime}$
together with these extra terms from $E^{\prime}$ degenerate after $E_{2}$.
Therefore, starting from the third layer spectral sequences $E_{\bullet}$ and
$F_{\bullet}$ are isomorphic and converge to the same thing
$\mathbf{k}[e_{1},\ldots,e_{2n-1}]$ as required.

As was discussed before, the only contribution to $E^{\prime\prime}_{2}$ comes
from diagrams $\gamma$ without flippable boxes and it is of the form
$\mathbf{k}[e_{1},\ldots,e_{2n-3},e^{\prime}_{1}]z_{\gamma}$. It is easy to
see that such diagram $\gamma$ can be obtained from some diagram $\beta$ with
$\mathrm{ht}(\beta)=n-1$ and $d(\beta,\lambda)=0$ by repeated application of
the $+$-operation. For every diagram $\lambda$ we put

$l(\lambda)=l_{n-1}(\lambda)=\sum_{i=1}^{n-1}\lambda_{i},$

and consider decreasing filtration of $E_{2}$ by $\mathcal{F}_{\geqslant
l}E_{2}$ consisting of all components of type $\beta$ with $l(\beta)\geqslant
l$. In order to establish the acyclicity of the subcomplex in question it is
enough to show acyclicity of the associated graded for this filtration. Next,
consider another grading on $E_{2}$ by the number of $e^{\prime}_{1}$ and the
induced grading on the associated graded space
$\mathop{\mathrm{gr}}_{\mathcal{F}}E_{2}$. This turns
$\mathop{\mathrm{gr}}_{\mathcal{F}}E_{2}$ into a collection of bicomplexes
where one differential acts on $e_{1}$ but preserves $e^{\prime}_{1}$, while
the second differential acts trivially on generators $e_{i}$ and
$d(e^{\prime}_{1}z_{\gamma})=z_{\gamma^{+}}$. Now, by taking cohomology with
respect to the latter differential we see that the associated graded of
$E^{\prime\prime}_{2}$ together with the extra terms from $E^{\prime}_{2}$
mentioned before is acyclic, and hence the $E^{\prime\prime}_{2}$ itself with
those extra terms is also acyclic.

This completes the proof of the proposition.

$\square$

######  Remark 2.3.

This proposition combined with the theorem 1.2 suggests that the only
contribution to the cohomology
$H^{\bullet}(\mathfrak{g},\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*}))$
comes from the indecomposable modules $\mathbf{k}$ and
$\mathop{\mathrm{missing}}{Ind}\nolimits_{\mathfrak{g}_{0}}^{\mathfrak{g}}(\mathbf{k})$.
Indeed, suppose we know that the invariant subspace
$\mathbf{k}\hookrightarrow\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$
is either a direct summand itself or a part of an indecomposable direct
summand isomorphic to
$\mathop{\mathrm{missing}}{Ind}\nolimits_{\mathfrak{g}_{0}}^{\mathfrak{g}}(\mathbf{k})$.
These two possibilities can be easily distinguished. First, observe that the
simple odd root of $\mathfrak{g}$ acts acyclically on
$\mathop{\mathrm{missing}}{Ind}\nolimits_{\mathfrak{g}_{0}}^{\mathfrak{g}}(\mathbf{k})$.
Next, write $V=X\oplus Y$, where $X$ is the standard representation of
$\mathfrak{gl}(n-1)$, and $Y$ is the standard representation of
$\mathfrak{gl}(1,1)$. Since $\Sigma^{\beta}(X)=0$ whenever
$\mathrm{ht}(\beta)>n-1$ we see that in the decomposition of
$\Sigma^{\lambda}(V)$ with $\mathrm{ht}(\lambda)>n-1$ every summand involves
at least one $Y$, and similarly for $\Sigma^{\lambda}(V^{*})$. Therefore, from
the representation theory of $\mathfrak{gl}(1,1)$-modules recalled above it
follows that the simple odd root of $\mathfrak{gl}(n,1)$ acts acyclically on
$\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$, which in
turn implies that it can not contain the trivial representation as a direct
summand. Then comparing theorem 1.2 and the proposition we conclude that
$\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$ with
$\mathrm{ht}(\lambda)\leqslant n-1$ has $\mathbf{k}$ as a direct summand, and
no summands other than $\mathbf{k}$ and
$\mathop{\mathrm{missing}}{Ind}\nolimits_{\mathfrak{g}_{0}}^{\mathfrak{g}}(\mathbf{k})$
can contribute to the cohomology.

It would be interesting to give an independent description of the structure of
the module
$\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$, as it
would provide alternative (and possibly more meaningful) proof of the
proposition 2.2.

##  3 Cohomology of $\mathcal{V}_{n,1}$

We begin this section with the standard opening argument used in calculation
of $H^{\bullet}(\mathcal{V}_{m,0},\mathbf{k})$ (for details we refer to
[Fuk]), which is also applicable in the general case of $\mathcal{V}_{m,n}$.

Consider Lie subalgebra $\mathfrak{gl}(m,n)\hookrightarrow\mathcal{V}_{m,n}$,
and let $V$ be the standard representation of $\mathfrak{gl}(m,n)$. We have
the associated spectral sequence

$E_{1}^{pq}=H^{q}(\mathfrak{gl}(m,n),\mathrm{Hom}(\Lambda^{p}(\mathcal{V}_{m,n}/\mathfrak{gl}(m,n)),\mathbf{k}))\Rightarrow
H^{p+q}(\mathcal{V}_{m,n},\mathbf{k}).$

As a $\mathfrak{gl}(m,n)$-module the associated graded
$\mathop{\mathrm{gr}}\mathcal{V}_{m,n}$ (recall that $\mathcal{V}_{m,n}$ is
the completed Lie superalgebra of vectorfields) is isomorphic to

$\mathop{\mathrm{gr}}\mathcal{V}_{m,n}\ \simeq\ \bigoplus_{n\geqslant
0}\left(S^{n}(V)\mathop{\otimes}\limits V^{*}\right).$

In this decomposition the Lie subalgebra
$\mathfrak{gl}(m,n)\hookrightarrow\mathcal{V}_{m,n}$ corresponds to the
summand $V\mathop{\otimes}\limits V^{*}$. Therefore, as a
$\mathfrak{gl}(m,n)$-module the space
$\mathrm{Hom}(\Lambda^{p}(\mathcal{V}_{m,n}/\mathfrak{gl}(m,n)),\mathbf{k})$
is isomorphic to

(3.0.1) $\bigoplus_{\sum
p_{i}=p}\Lambda^{p_{i}}\left(S^{i}(V^{*})\mathop{\otimes}\limits V\right),$

where $i\geqslant 0$ and $i\neq 1$. In order for the cohomology to be non-zero
we need equal number of instances of $V$ and $V^{*}$ in the above expression,
which leads to

$\sum_{i\geqslant 0,i\neq 1}p_{i}=\sum_{i\geqslant 0,i\neq 1}ip_{i},$

or equivalently, $p_{0}=\sum\limits_{i\geqslant 2}(i-1)p_{i}$.

On the other hand, as we discussed in 1.1 the contribution to cohomology only
comes from terms
$\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$, and the
latter contains one-dimensional invariant subspace. Due to the symmetry of
$S^{i}(V^{*})$ in (3.0.1) in order to obtain an invariant we must have

$p_{0}\leqslant\sum_{i\geqslant 2}p_{i}.$

Combining the two conditions we find that $p_{i}=0$ for $i\geqslant 3$ and
$p_{0}=p_{2}$. In other words, it is enough to focus our attention on the
space
$\Lambda^{p}(V)\mathop{\otimes}\limits\Lambda^{p}(S^{2}(V^{*})\mathop{\otimes}\limits
V)$.

######  Lemma 3.1.

Let $\mathfrak{g}=\mathfrak{gl}(m,n)$, and $V$ the standard representation of
$\mathfrak{g}$, then

$H^{\bullet}(\mathfrak{g},\Lambda^{p}(V)\mathop{\otimes}\limits\Lambda^{p}(S^{2}(V^{*})\mathop{\otimes}\limits
V))\ \simeq\
\bigoplus_{|\lambda|=p}H^{\bullet}(\mathfrak{g},\Sigma^{\widetilde{\lambda}}(V)\mathop{\otimes}\limits\Sigma^{\widetilde{\lambda}}(V^{*})),$

where $\widetilde{\lambda}$ is obtained from $\lambda$ by shifting it to the
right and adding first column of height $p$. In other words,
$\widetilde{\lambda}$ is a diagram of height $p$, such that
$\widetilde{\lambda}_{i}=\lambda_{i}+1$ for $1\leqslant i\leqslant p$.

Proof: First, recall that we have

$\Lambda^{p}(S^{2}(V^{*})\mathop{\otimes}\limits V)\ \simeq\
\bigoplus_{|\lambda|=p}\Sigma^{\lambda}(S^{2}(V^{*}))\mathop{\otimes}\limits\Sigma^{\lambda^{\prime}}(V).$

Since the contribution to the cohomology comes only from terms of the form
$\Sigma^{\alpha}(V)\mathop{\otimes}\limits\Sigma^{\alpha}(V^{*})$, we can
disregard everything else. Observe that the product
$\Lambda^{p}(V)\mathop{\otimes}\limits\Sigma^{\lambda^{\prime}}(V)$ is the sum
of $\Sigma^{\beta}(V)$ with $\mathrm{ht}(\beta)\geqslant p$. On the other
hand, the height of diagrams appearing in $\Sigma^{\lambda}(S^{2}(V^{*}))$ is
less or equal than $p$. Hence, it is enough to consider only diagrams of
height $p$.

Notice, that in the product
$\Lambda^{p}(V)\mathop{\otimes}\limits\Sigma^{\lambda^{\prime}}(V)$ there is
only one such diagram, namely $\widetilde{\lambda^{\prime}}$. Furthermore, in
constructing diagrams in $\Sigma^{\lambda}(S^{2}(V^{*}))$ the first box of
every pair must go into the first column, and since they anticommute with each
other the commutation relation for the rest of the boxes are opposite to that
of $\lambda$, i.e. they form the transposed diagram $\lambda^{\prime}$. Thus,
we have a term $\Sigma^{\widetilde{\lambda^{\prime}}}(V^{*})$ in the expansion
of $\Sigma^{\lambda}(S^{2}(V^{*}))$. This completes the proof of the lemma.

$\square$

######  Remark 3.2.

From the lemma it is immediate to see what invariants can appear in

$\Lambda^{\bullet}(V)\mathop{\otimes}\limits\Lambda^{\bullet}(S^{2}(V^{*})\mathop{\otimes}\limits
V)$

and give the relations for the multiplicative generators. Specifically, the
diagram $\widetilde{\lambda}$ must be contained in a thick hook with $m$ rows
and $n$ columns (see 1).

As an example, we illustrate it in the cases $\mathcal{V}_{m,0}$ and
$\mathcal{V}_{0,n}$, that have been previous handled using a different
argument in [Fuk] and [AF]. In the case $\mathcal{V}_{m,0}$ we have
$\mathrm{ht}(\widetilde{\lambda})\leqslant m$, therefore the space of
invariants is generated by Chern classes $c_{i}$ of degree $2i$, for
$1\leqslant i\leqslant m$ (corresponding to columns of height $i$), with the
condition that the total degree is less or equal than $2m$ (the height of the
first column).

For $\mathcal{V}_{0,n}$ the diagram $\widetilde{\lambda}$ must be contained in
the first $n$ columns, so the space of invariants is generated by the Segre
classes $s_{i}$ of degree $2i$, for $1\leqslant i\leqslant n-1$ with no
relations. Here we also notice that since $s_{i}\equiv c_{i}$ modulo
subalgebra generated by all the lower degree classes (and up to a coefficient
that is of no importance in characteristic $0$) the classes $s_{i}$ and
$c_{i}$ are often interchangeable when studying the differentials in the
spectral sequence.

From this point forward we restrict the discussion to $\mathcal{V}_{n,1}$.
From the previous lemma we see that the space of invariants in
$\Lambda^{\bullet}(V)\mathop{\otimes}\limits\Lambda^{\bullet}(S^{2}(V^{*})\mathop{\otimes}\limits
V)$ is generated by the classes $c_{i}$ of degree $2i$, for $1\leqslant
i\leqslant n$, with no relations imposed on them.

Let $BGL(n)$ be the classifying space of the topological group
$GL(n,\mathbb{C})$ and

$p\colon EGL(n)\to BGL(n)$

be the tautological principal $GL(n)$-bundle. Recall that the de Rham
cohomology

$H^{\bullet}(BGL(n),\mathbf{k})\simeq\mathbf{k}[c_{1},c_{2},\ldots,c_{n}],$

with generators $c_{i}$ of degree $2i$. Consider $2d$-dimensional skeleton
$\mathop{\mathrm{sk}}_{2d}BGL(n)\hookrightarrow BGL(n)$ and denote
$X_{2d}=p^{-1}(\mathop{\mathrm{sk}}_{2d}BGL(n))$. We will also denote by $\SS$
the topological suspension functor. Now we are ready to prove the main theorem
of this paper.

######  Theorem 3.3.

We have an isomorphism

$H^{\bullet}(\mathcal{V}_{n,1},\mathbf{k})\ \simeq\
H^{\bullet}(\SS^{2}X_{2(n-1)},\mathbf{k}).$

Proof: Consider the spectral sequence associated to the Lie subalgebra
$\mathfrak{gl}(n,1)\hookrightarrow\mathcal{V}_{n,1}$. From the discussion
above and lemma 3.1 we find that $E_{1}^{pq}=0$ if $p$ is odd and

$E_{1}^{2p,q}=\bigoplus_{|\lambda|=p}H^{\bullet}(\mathfrak{gl}(n,1),\Sigma^{\widetilde{\lambda}}(V)\mathop{\otimes}\limits\Sigma^{\widetilde{\lambda}}(V^{*})).$

Next, from proposition 2.2 we see that

$E_{1}^{2p,q}=\begin{cases}\bigoplus_{|\lambda|=p}\mathbf{k}[e_{1},\ldots,e_{2n-1}],&\text{if
$p\leqslant n-1$,}\\\
\bigoplus_{|\lambda|=p}\mathbf{k}[e_{1},\ldots,e_{2n-1},e^{\prime}_{1}],&\text{if
$p\geqslant n$.}\end{cases}$

We will also consider the spectral sequence for the cohomology of the total
space of the tautological bundle $EGL(n)\to BGL(n)$ with coefficients in
$\mathbf{k}[e^{\prime}_{1}]$

$F_{1}^{\bullet\bullet}=\mathbf{k}[e_{1},\ldots,e_{2n-1}]\mathop{\otimes}\limits\mathbf{k}[c_{1},\ldots,c_{n}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{1}]\Rightarrow
H^{\bullet}(EGL(n),\mathbf{k}[e^{\prime}_{1}])\simeq\mathbf{k}[e^{\prime}_{1}],$

since the $EGL(n)$ is contractible.

Let us look at the short exact sequence $E_{1}\hookrightarrow
F_{1}\twoheadrightarrow E_{1}/F_{1}$. For the quotient we have

$(E_{1}/F_{1})^{\bullet\bullet}=\left(\mathbf{k}[e_{1},\ldots,e_{2n-1}]\mathop{\otimes}\limits\mathbf{k}[c_{1},\ldots,c_{n}]/I_{n}\right)e^{\prime}_{1},$

where $I_{n}$ is spanned by all the monomials of total degree at least $2n$.
The standard argument shows that it converges to
$H^{\bullet}(X_{2(n-1)},\mathbf{k})\cdot e^{\prime}_{1}$. Now from the long
exact sequence of cohomology associated with this triple of spectral sequences
we find that

$\displaystyle H^{0}(\mathcal{V}_{n,1},\mathbf{k})$
$\displaystyle=\mathbf{k},$ $\displaystyle
H^{1}(\mathcal{V}_{n,1},\mathbf{k})$
$\displaystyle=H^{2}(\mathcal{V}_{n,1},\mathbf{k})=0,$ $\displaystyle
H^{i}(\mathcal{V}_{n,1},\mathbf{k})$ $\displaystyle\simeq
H^{i-2}(X_{2(n-1)},\mathbf{k}),\quad\text{for $i>2$}.$

$\square$

######  Remark 3.4 (Cohomology of $\mathcal{V}_{1,n}$.).

As another application of the methodology developed here we give an
alternative proof of

$H^{\bullet}(\mathcal{V}_{1,n},\mathbf{k})\ \simeq\
H^{\bullet}(S^{2n-1},\mathbf{k}),$

which is of course a special case of theorem  ‣ 0 Introduction.

As before we write $V$ for the standard representation of $\mathfrak{gl}(n,1)$
and let $W$ be the standard representation of $\mathfrak{gl}(1,n)$. Notice
that they differ by the change of parity, i.e. $W=\Pi V$. Therefore, up to
parity we have isomorphisms
$\Sigma^{\lambda}(W)\simeq\Sigma^{\lambda^{\prime}}(V)$. Now, from lemma 3.1
and the discussion in remark 3.2 we find that diagrams $\lambda$ contributing
to the first layer of the spectral sequence are contained in the thick hook
with $1$ row and $n$ columns. Consider a free algebra generated by elements
$s_{i}$ of degree $2i$, for all $i\geqslant 1$ and the ideal $J=(s_{j}\mid
j\geqslant n)$. Then we see that the space of invariants can be described as

$\left(\Lambda^{\bullet}(W)\mathop{\otimes}\limits\Lambda^{\bullet}(S^{2}(W^{*})\mathop{\otimes}\limits
W)\right)^{\mathfrak{gl}(1,n)}\ \simeq\
\mathbf{k}[s_{1},s_{2},\ldots]/(J)^{2}.$

Denote by $z_{\lambda}$ the basis vector in the first layer of the spectral
sequence coming from
$\Sigma^{\lambda}(W)\mathop{\otimes}\limits\Sigma^{\lambda}(W^{*})$. From
proposition 2.2 and the relation between $W$ and $V$ mentioned above we see
that elements $e^{\prime}_{1}$ appear only for $z_{\lambda}$ with
$\mathrm{ht}(\lambda^{\prime})\geqslant n$. Proceeding as in the proof of
proposition 2.2 we consider decreasing filtration $\mathcal{F}_{l}E_{2}$ by
$l_{n}(\lambda^{\prime})$ (see 2.2 for the notation) and another grading on
$E_{2}$ by the number of $e^{\prime}_{1}$, which makes it a collection of
bicomplexes. After passing to the associated graded one of the differentials
in the bicomplex can be written as
$d(e^{\prime}_{1}z_{\lambda})=z_{\lambda_{+}}$, where
$\lambda_{+}=((\lambda^{\prime})^{+})^{\prime}$, i.e. obtained from $\lambda$
by adding one box in the first row. Taking cohomology with respect to this
differential (and keeping the other differential intact) we obtain a spectral
sequence which starting from the second layer is isomorphic to another
spectral sequence $F_{\bullet}$, that starts with the first layer isomorphic
to

$F_{1}\simeq\mathbf{k}[e_{1},\ldots,e_{2n-1}]\mathop{\otimes}\limits\mathbf{k}[s_{1},\ldots,s_{n-1}].$

The elements $e_{2i-1}$ for $1\leqslant i<n$ are transgressive, i.e. they are
killed by differentials of the spectral sequence, hence we find

$H^{\bullet}(\mathcal{V}_{1,n},\mathbf{k})\ \simeq\ \mathbf{k}[e_{2n-1}].$

$\square$

## References

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# On the Soundness of XAI in Prognostics and Health Management (PHM)

David Solís-Martín, Juan Galán-Páez
Departamento de Ciencias de la Computación e Inteligencia Artificial
Datrik Intelligence, S.A.
Universidad de Sevilla
Sevilla
{david.solis<EMAIL_ADDRESS>
{dsolis<EMAIL_ADDRESS>
&Joaquín Borrego-Díaz
Universidad de Sevilla
Sevilla
<EMAIL_ADDRESS>

###### Abstract

The aim of Predictive Maintenance, within the field of Prognostics and Health
Management (PHM), is to identify and anticipate potential issues in the
equipment before these become critical. The main challenge to be addressed is
to assess the amount of time a piece of equipment will function effectively
before it fails, which is known as Remaining Useful Life (RUL). Deep Learning
(DL) models, such as Deep Convolutional Neural Networks (DCNN) and Long Short-
Term Memory (LSTM) networks, have been widely adopted to address the task,
with great success. However, it is well known that this kind of black box
models are opaque decision systems, and it may be hard to explain its outputs
to stakeholders (experts in the industrial equipment). Due to the large number
of parameters that determine the behavior of these complex models,
understanding the reasoning behind the predictions is challenging. This work
presents a critical and comparative revision on a number of XAI methods
applied on time series regression model for PM. The aim is to explore XAI
methods within time series regression, which have been less studied than those
for time series classification. The model used during the experimentation is a
DCNN trained to predict the RUL of an aircraft engine. The methods are
reviewed and compared using a set of metrics that quantifies a number of
desirable properties that any XAI method should fulfill. The results show that
GRAD-CAM is the most robust method, and that the best layer is not the bottom
one, as is commonly seen within the context of Image Processing.

_Keywords_ XAI; Interpretability; Predictive Maintenance; Prognostics and
Health Management; Remaining Useful Life; Deep Learning; Convolutional Neural
Network; Comparative Analysis; XAI Evaluation and Accountability

##  1 Introduction

Incipient AI systems, as small decision trees, were straightly interpretable
but had limited capabilities. Nevertheless, during the last years, the notable
increase in the performance of predictive models (for both classification and
regression) has been accompanied by an increase in model complexity. This has
been at the expense of losing the understanding capacity of the reasons behind
each particular prediction. This kind of model is known as _black-box_
[Pomerleau(1993)] due to the opaqueness of its behavior. Such obscurity
becomes a problem, especially when the predictions of a model impact different
dimensions within the human realm (such as Medicine, Law, Profiling,
autonomous driving, or Defense, among others) [Goodman(2017)]. It is also
important to note that opaque models are difficult to debug, as opposed to the
interpretable ones, which facilitate the detection of the source of its
errors/bias and the implementation of a solution [Gilpin(2018)].

###  1.1 Explainable Artificial Intelligence

Explainable AI (XAI) addresses these issues by proposing machine learning (ML)
techniques that generate explanations of black-box models or create more
transparent models (in particular for _post-hoc_ explainability).
[Arrieta(2020)]. Post-hoc explainability techniques can be divided into model-
agnostic and model-specific techniques. Model-agnostic techniques encompass
those that can be applied to any ML model, such as LIME [Ribeiro(2016)] or
SHAP [Lundberg(2017)] (SHapley Additive exPlanations), for example. Whereas
model-specific techniques are designed for certain ML models, such as GRAD-CAM
(Gradient-weighted Class Activation Mapping) [Selvaraju(2017)], Saliency Maps
[Simonyan(2013)], or Layer-wise Relevance Propagation (LRP) [Bach(2015)],
which are focused on deep learning (DL) models.

Regarding the kind of tasks where XAI can be applied, it is common to find
applications in classification tasks with tabular and image data, while
regression tasks -signal processing, among others- have received little
attention. The higher number of works devoted to XAI for classification tasks
is due to the ease of its application, since implicit knowledge exists around
each class [Letzgus(2021)]. Similarly, there are not many works on XAI applied
to time series models [Schlegel(2019)]. The non-intuitive nature of time
series [Siddiqui(2019)] makes them harder to be understood. Even nowadays, to
the best of our knowledge, it seems that there is no specific XAI method
particularly successful for this kind of data.

In the same way that it does not exist a model best suited to solve any ML
task, there is no particular XAI method that will provide the best explanation
of any model. A significant body of literature devoted to innovations in novel
interpretable models and explanation strategies can be found. However,
quantifying the correctness of their explanations remains challenging. Most ML
interpretability research efforts are not aimed at comparing the explanation
quality (measuring it) provided by XAI methods [Carvalho(2019)]
[Vollert(2021)]. It can find two types of _indicators_ for the assessment and
comparison of explanations: qualitative and quantitative. Quantitative
indicators, which are the focus of this paper, are designed to measure
desirable characteristics that any XAI method should have. The metrics
approximate the level of accomplishment of each characteristic, thus allowing
us to measure them on any XAI method. As these metrics are a form to estimate
the accomplishment level, they will be referred to as _proxies_. Numerical
proxies are useful to assess the explanation quality, providing a
straightforward way to compare different explanations.

A (non-exhaustive) list of works on proxies show its usefulness. In
[Schlegel(2019)], Schlegel, et al. apply several XAI methods, usually used
with models built from image and text data, and propose a methodology to
evaluate them on time series. Samek, at al. [Samek(2017)] apply a perturbation
method on the variables that are important in the prediction generation, to
measure the quality of the explanation. The works of Doshi-Velez, et al. and
Honegger [Doshi(2017)] [Honegger(2018)] propose three proxies (called axioms
in those works) for measuring the consistency of explanation methods. The
three proxies are identity (identical samples must have identical
explanations), separability (non-identical samples cannot have identical
explanations) and stability (similar samples must have similar explanations).
There exist other works that propose different proxies for more specific
models or tasks [Silva(2018)].

###  1.2 XAI and Predictive Maintenance

The industrial maintenance process consists of three main stages. The first
stage involves identifying and characterizing any faults that have occurred in
the system. In the second stage, known as the diagnosis phase, the internal
location of the defects is determined, including which parts of the system are
affected and what may have caused the faults. In the final stage, known as the
prognosis phase, the gathered information is used to predict the machine’s
operating state, or Remaining Useful Life (RUL) at any given time, based on a
description of each system part and its condition. The first two are
classification problems, while prognosis is commonly addressed as a regression
problem. In the field of predictive maintenance (Pdm), there is an important
lack of XAI methods for industrial prognosis problems [Vollert(2021)] and it
is difficult to find existing works in which new XAI methods are developed, or
existing ones are applied within this context. Hong, et al. [Hong(2020)] use
SHAP to explain predictions in RUL prediction. Similarly, Szelazek, et al.
[Szelazek(2020)] use an adaptation of SHAP for decision trees applied to steel
production systems prognosis, to predict when the thickness of the steel is
out of specifications. Serranilla, et al. [Serradilla(2021)] apply LIME to
models used in the estimation of bushings remaining time of life. Recently,
Ferrano, et al. [Ferrano(2022)] have applied SHAP and LIME in hard disk drive
failure prognosis.

###  1.3 Aim and structure of the paper

Raw signal time series are frequently voluminous and challenging to analyze
data. Due to this issue, a quantitative method must verify the quality of
explanations [Schlegel(2019)]. This paper considers five XAI methods to
address regression tasks on signal time series; specifically, for system
prognosis within the context of PHM. Two of them, SHAP and LIME, are model-
agnostic XAI methods. The other three (layer-wise relevance propagation,
gradient activation mapping, and saliency maps) are neural network-specific.
It is worth noting that all these methods have been adapted in this paper to
work with time series regression models.

This article aims to present several contributions to the field of Explainable
Artificial Intelligence. Firstly, it presents a comprehensive review of eight
existing proxy methods for evaluating the interpretability of machine learning
models. Secondly, it proposes a novel proxy to measure the time-dependence of
XAI methods, which has not been previously explored in the literature.
Thirdly, an alternative version of Grad-CAM is proposed, which takes into
account both the time and time series dimensions of the input, improving its
interpretability. Finally, the importance of layers in Grad-CAM for
explainability is evaluated using the proposed proxies.

The article is organized into five main sections. Section 2 covers the
Materials and Methods used in the study, including the XAI methods and the
perturbation and neighborhood techniques. Section 2.3 focuses on the
validation of the XAI method explanations using quantitative proxies. Section
3 provides details on the experiments conducted in the study, including the
dataset and the black-box model used. Section 3.2 describes the experiments
themselves, and Section 3.3 presents the results of the study. Finally,
Section 4 provides a discussion of the findings and their implications.

##  2 Materials and Methods

This section describes the different XAI methods under study (Subsect 2.1), as
well as the consistency of explanation proxies, used in the experiments
carried out (Subsect 2.2).

###  2.1 XAI methods

This section provides a brief description of the XAI models used in the
experiments.

####  2.1.1 Local interpretable model-agnostic explanations

Local Interpretable Model-agnostic Explanations (LIME) [Ribeiro(2016)] is a
XAI method based on a surrogate model. In XAI, surrogate models are trained to
approximate the predictions of the black-box model. These models would be
white-box models, easily interpreted (sparse linear models or simple decision
trees). In the case of LIME, the surrogate model is trained to approximate an
individual prediction and the predictions of its neighborhood obtained by
perturbing the individual sample studied. The LIME surrogate model is trained
with a data representation of the original sample $x\>\epsilon\>R^{d}$. The
representation uses $x^{\prime}\>\epsilon\\{0,1\\}^{d^{\prime}}$ to state the
non-perturbation/perturbation of each original feature. Mathematically, the
explanations obtained with LIME can be expressed as:

$\xi(x)=\underset{g\>\epsilon\>G}{argmin}\>\mathcal{L}(f,g,\pi_{x})+\Omega(g)$
(1)

where $g$ is a surrogate model from the class $G$ of the all interpretable
models. The component $\Omega(g)$ is used as regularization to keep the
complexity of $g$ low, since high complexity is opposed to the
interpretability concept. The model being explained is denoted as $f$ and
$\mathcal{L}$ determines the performance of $g$ fitting the locality defined
by $\pi$ as a proximity measurement function and $\pi_{x}=\pi(x,\cdot)$.
Finally, each training sample is weighted with the distance between the
perturbed sample and the original sample.

####  2.1.2 SHapley Additive exPlanations

SHapley Additive exPlanations (SHAP) [Lundberg(2017)], is also a method to
explain individual predictions, as LIME. The SHAP method explains each feature
by computing shapely values from coalitional game theory. Shapley value can be
described as the expected average of a player’s marginal contribution (by
considering all possible combinations). It enables the determination of a
payoff for all players, even when each one has contributed differently. In
SHAP, each feature is considered a player. Thus, the coalition vector
$x^{\prime}$, or simplified features vector, is composed of 1/0’s representing
the presence or absence of a feature, respectively. The contribution of each
feature $\phi_{i}$ is estimated based on its _marginal contribution_
[Shapley(1953)], and computed as follows:

$\phi_{i}(f,x)=\underset{z^{\prime}\>\subseteq\>x^{\prime}}{\sum}\frac{\left|z^{\prime}\right|!(M-\left|z^{\prime}\right|-1)!}{M!}\left[f_{x}(z^{\prime})-f_{x}(z^{\prime}\setminus
i\right)]$ (2)

where $\left|z^{\prime}\right|$ is the number of non-zero entries in
$z^{\prime}$, and $z^{\prime}\subseteq x^{\prime}$ represents all vectors
where the non-zero entries are a subset of the coalition vector $x^{\prime}$.
The values $\phi_{i}$ are known as Shapely values, and it has been
demonstrated that they satisfy the properties of _local accuracy, missingness,
and consistency_ [Lundberg(2017)]. Based on these contribution values, a
linear model is defined to obtain the explainable model $g$:

$g(x^{\prime})=\phi_{o}+\underset{j=1}{\overset{M}{\sum}}\phi_{j}$ (3)

The explainable model $g$ is optimized by minimizing the mean squared error
between $g$ and the predictions over the perturbed samples
$f(h_{x}(z^{\prime}))$. The main difference with LIME is that in SHAP, each
sample is weighted based on the number of 1’s in $z^{\prime}$. The weighting
function, called the weighting kernel, gives rise to the so-called Kernel SHAP
method. The formula for the weight $\pi_{x}(z^{\prime})$ is given by:

$\pi_{x}(z^{\prime})=\frac{(M-1)}{\binom{M}{\left|z^{\prime}\right|}(M-\left|z^{\prime}\right|)}$
(4)

The intuition behind this is that isolating features provides more information
about their contribution to the prediction. This approach computes only the
more informative coalitions, as computing all possible combinations is an
intractable problem in most cases.

####  2.1.3 Layer-Wise Propagation

The Layer-wise Propagation (LRP) method [Bach(2015)] aims to interpret the
predictions of deep neural networks. It is thus a model-specific XAI method.
The goal is to attribute relevance scores to each input feature (or neuron) of
a neural network, indicating its contribution to the final prediction. LRP
works by propagating relevance scores from the output layer of the network
back to its input layer. The relevance scores are initialized at the output
layer: a score of 1 is assigned to the neuron corresponding to the predicted
class and 0 to all others. Then, relevance is propagated backward from layer
to layer using a propagation rule that distributes the relevance scores among
the inputs of each neuron in proportion to their contribution to the neuron’s
output. The rule ensures that the sum of relevance scores at each layer is
conserved. The propagation rule is defined by the equation.

$R_{i}=\underset{k}{\sum}\frac{z_{ik}}{\sum_{0,j}z_{jk}}R_{k}$ (5)

where $R$ represents the propagation relevance score, $j$ and $k$ refer to
neurons in two consecutive layers, and $z_{jk}=a_{j}w_{jk}$ denotes how neuron
$j$ influences the relevance of neuron $k$ based on the activation of each
neuron. The denominator enforces the conservation property.

####  2.1.4 Image-Specific Class Saliency

Image-Specific Class Saliency [Simonyan(2013)] is one of the earliest pixel
attribution methods existing in the literature. Pixel attribution methods aim
to explain the contribution of each individual pixel, within an image, to the
model’s output. These methods are typically used in computer vision tasks such
as image classification or object detection. However, in this work,
attribution is assigned to each element of each time series, rather than
individual pixels in an image. It is based on approximating the scoring or
loss function, $S_{c}(x)$, with a linear relationship in the neighborhood of
x:

$S_{c}(x)\approx w_{c}^{T}x+b$ (6)

where each element of $w_{c}$ is the importance of the corresponding element
in $x$. The $w_{c}$ vector of importance values is computed via the derivative
of $S_{c}$ with respect to the input $x$:

$w=\frac{\partial S_{c}}{\partial x}\left.\right|_{x_{0}}$ (7)

This method was originally designed to work in Image Processing with neural
networks, hence each element of $w$ is associated with the importance of each
pixel.

####  2.1.5 Gradient-weighted Class Activation Mapping

Finally, Gradient-weighted Class Activation Mapping (Grad-CAM) generalizes CAM
[Zhou(2016)], which determines the significance of each neuron in the
prediction by considering the gradient information that flows into the last
convolutional layer of the CNN. Grad-CAM computes the gradient $y^{c}$ of
class $c$ with respect to a feature map $A^{k}$ of a convolutional layer,
which is then globally averaged, obtaining the neuron importance
$\alpha_{k}^{c}$ of the feature map $A^{k}$:

$\alpha_{k}^{c}=\overset{global\>average\>pooling}{\overbrace{\frac{1}{Z}\underset{i}{\sum}\underset{j}{\sum}}}\underset{gradients\>via\>backprop}{\underbrace{\frac{\partial
y^{c}}{\partial A^{k}}}}$ (8)

After computing the importance of all feature maps, a heat map can be obtained
through a weighted combination of them. The authors apply a ReLU activation
since they are only interested in positive importance;

$L^{c}=ReLU\left(\underset{k}{\sum}{\alpha_{k}^{c}A^{k}}\right)$ (9)

Unlike saliency maps, GRAD-CAM associates importance by regions of the input.
The size of them depends on the size of the convolutional layer. Usually,
interpolation is applied to the original heat map to expand it to the overall
size of the input.

As this work is focused on time series, we propose introducing additional
elements to Grad-CAM with the aim of exploiting the possible stationary
information of the signal. This is achieved by introducing an additional
component to the Grad-CAM heat map calculation, namely the time component
contribution. Moreover, a second component was introduced to exploit the
importance each time series has in a multivariate time series problem. Thus,
the final Grad-CAM attribution equation read as follows, namely the individual
time series contribution:

$\alpha_{k}^{c}=\overset{individual\>feature\>contribution}{\overbrace{\frac{1}{T*F}\overset{T}{\underset{i}{\sum}}\overset{F}{\underset{j}{\sum}}\frac{\partial
y^{c}}{\partial
A^{k}}}}+\underset{time\>component\>contribution}{\beta\underbrace{\frac{1}{T}\overset{T}{\underset{i}{\sum}}\frac{\partial
y^{c}}{\partial
A^{k}}}}+\overset{individual\>time\>series\>contribution}{\overbrace{\sigma\frac{1}{F}\overset{F}{\underset{j}{\sum}}\frac{\partial
y^{c}}{\partial A^{k}}}}$ (10)

where $T$ and $F$ are the time units present in the time series, and the
number of time series, respectively. The components $\beta$ and $\sigma$ are
used to weigh these two new components.

Figure 1 displays examples of heat maps generated by each method. Each heat
map is a matrix of 20 by 160 values, representing 20 time series and 160 time
units, where a relative importance is assigned to each item in the time
series.

Figure 1: Heat maps generated for each of the five tested methods

###  2.2 Perturbation and neighborhood

The LRP, saliency map, and GRAD-CAM techniques can be directly used on time
series data. However, LIME and SHAP assume that significant changes in the
performance of a well-trained model will occur if its relevant features (time
points) are altered. Due to the high dimensionality of time series inputs, in
order to achieve the former, it is necessary to specify a grouping of time
series elements to analyze the impact on each group instead of on single time
points. In image processing, this is achieved through super-pixels, which are
groups of connected pixels that share a common characteristic.

In this work, time series are segmented considering adjacent elements. Two
different segmentation approaches are used. The first one is called uniform
segmentation, which is the most basic method, and involves splitting the time
series $ts={t_{0},t_{1},t_{2},...,t_{n}}$ into equally sized windows without
overlapping. The total number of windows is $d=\frac{n}{m}$, where $m$ is the
size of the window. If $n$ is not divisible by $m$, the last window may be
adjusted. The second segmentation minimizes the sum of $l2$ errors by grouping
time points as follows:

$\epsilon_{l2}(ts_{i,j})=\overset{j}{\underset{k=i}{\sum}}\left|ts_{k}-\overline{ts}_{i,j}\right|^{2}$
(11)

where $ts_{i,j}$ represents a segment of the time series signal $ts$ that goes
from element $i$ to element $j$, and $\overline{ts_{i,j}}$ is the mean of that
segment. That is to say, the final cost of the segmentation is the sum of the
$l2$ errors, calculated between each pair of adjacent elements within the
segment. To find the optimal segmentation, a dynamic programming approach is
employed, similarly to that described in [Truong(2020)]. The two segmentation
strategies are shown in figure 2.

Figure 2: Segmentation approaches: Left, uniform segmentation. Right, minimal
error segmentation

Once the segmentation is complete, a perturbation method must be applied to
create the neighborhood of the time series for SHAP and LIME. The following
five perturbation techniques have been applied on the segmented time series,
in the different experiments carried out:

  * •

Zero: The values in $ts_{i,j}$ are set to zero.

  * •

One: The values in $ts_{i,j}$ are set to one.

  * •

Mean: The values in $ts_{i,j}$ are replaced with the mean of that segment
($\overline{ts}_{i,j}$).

  * •

Uniform Noise: The values in $ts_{i,j}$ are replaced with random noise
following a uniform distribution between the minimum and maximum values of the
feature.

  * •

Normal Noise: The values in $ts_{i,j}$ are replaced with random noise
following a normal distribution with mean and standard deviation of the
feature.

To obtain a perturbed sample $x^{\prime}$, firstly, it is divided into $n$
segments. Then, from this segmentation, a binary representation $z^{\prime}$,
identifying which segments will be perturbed, is randomly generated:

$x^{\prime}=h(x,z^{\prime})=\left(h(x,z^{\prime})_{1},h(x,z^{\prime})_{2},...,h(x,z^{\prime})_{n}\right)$
(12)

where

$h(x,z^{\prime})_{i}=\left\\{\begin{matrix}g(x,i)&if\>z^{\prime}_{i}\>is\>equal\>0\\\
p(g(x,i))&if\>z^{\prime}_{i}\>is\>equal\>1\end{matrix}\right.\qquad\qquad
i\in\\{1,...,n\\}$ (13)

with $p$ being a perturbation function and $g$ a segmentation function.

###  2.3 Validation of XAI method explanations

This study uses the most relevant quantitative proxies found in the literature
[Honegger(2018), Rokade(2008)], to evaluate and compare each method. Different
methodologies need to be followed depending on the proxy used to evaluate the
interpretability of machine learning models. These methodologies are depicted
graphically in Figure 3. Approach A involves using different samples and their
corresponding explanations to compute metrics such as identity, separability,
and stability. In approach B, a specific sample is perturbed using its own
explanation, and the difference in prediction errors (between the original and
perturbed sample) is computed. This methodology is used to evaluate metrics
such as selectivity, coherence, correctness, and congruency. Approach C
involves using the explanation of the original sample to build the perturbed
sample, then the explanations from both the original and perturbed samples are
used to compute the acumen proxy.

Figure 3: Methodologies can be used to compute proxies for evaluating the
interpretability of machine learning models.

The following are the desirable characteristics that each XAI method should
accomplish, and the proxy used for each one in this work:

  * •

_Identity_ : The principle of identity states that identical objects should
receive identical explanations. This estimates level of intrinsic non-
determinism in the method.

$\forall a,b\>\left(d(x_{a},x_{b})=0\implies
d(\epsilon_{a},\epsilon_{b})=0\right)$ (14)

$x$ are samples, $d$ is a distance function and $\epsilon$ explanation vectors
(which explain the prediction of each sample).

  * •

_Separability_ : Non-identical objects cannot have identical explanations.

$\forall a,b\>\left(d(x_{a},x_{b})\neq 0\implies
d(\epsilon_{a},\epsilon_{b})>0\right)$ (15)

If a feature is not actually needed for the prediction, then two samples that
differ only in that feature will have the same prediction. In this scenario,
the explanation method could provide the same explanation, even though the
samples are different. For the sake of simplicity, this proxy is based on the
assumption that every feature has a minimum level of importance, positive or
negative, in the predictions.

  * •

_Stability_ : Similar objects must have similar explanations. This is built on
the idea that an explanation method should only return similar explanations
for slightly different objects. The Spearman correlation $\rho$ is used to
define this:

$\rho(\left\\{d(x_{i},x_{0}),d(x_{i},x_{1}),...,d(x_{i},x_{n})\right\\},\left\\{d(\epsilon_{i},\epsilon_{0}),d(\epsilon_{i},\epsilon_{1}),...,d(\epsilon_{i},\epsilon_{n})\right\\})\underset{\forall
i}{=}\rho_{i}>0$ (16)

  * •

_Selectivity_. The elimination of relevant variables must affect negatively to
the prediction [Bach(2015)] [Samek(2016)]. To compute the selectivity, the
features are ordered from the most to least relevant. One by one the features
are removed, by setting it to zero for example, and the residual errors are
obtained to get the area under the curve (AUC).

  * •

_Coherence_. It computes the difference between the prediction error
$p_{e}^{i}$ over the original signal and the prediction error $e_{e}^{i}$ of a
new signal where the non-important features are removed.

$\alpha_{i}=\left|p_{e}^{i}-e_{e}^{i}\right|$ (17)

where $\alpha_{i}$ is the coherence of a sample.

  * •

_Completeness_. It evaluates the percentage of the explanation error from its
respective prediction error.

$\gamma_{i}=\frac{e_{e}^{i}}{p_{e}^{i}}$ (18)

  * •

_Congruency_. The standard deviation of the coherence provides the congruency
proxy. This metric help to capture the variability of the coherence.

$\delta=\sqrt{\frac{\sum((\alpha_{i}-\overline{\alpha})^{2}}{N}}$ (19)

where $\overline{\alpha}$ the average coherence over a set of $N$ samples:

$\overline{\alpha}=\frac{\sum\alpha_{i}}{N}$ (20)

  * •

_Acumen_. It is a new proxy proposed by the authors for the first time in this
paper, based on the idea that an important feature according to the XAI method
should be one of the least important after it is perturbed. This proxy aims to
detect whether the XAI method depends on the position of the feature, in our
case, the time dimension. It is computed by comparing the ranking position of
each important feature after perturbing it.

$\varpi=1-\frac{\underset{f_{i}\in\mathcal{I}}{\sum}\frac{p_{a}(f_{i})}{N}}{M}$
(21)

where $\mathcal{I}$ is the set of $M$ important features before the
perturbation, $p_{a}(f_{i})$ is a function that returns the position of
feature $f_{i}$ within the importances vector after the perturbation, where
features with lower importance are located at the beginning of the vector.

Some of the previously depicted methods for evaluating the interpretability of
machine learning models perturb the most important features identified by the
XAI method. In our work, we define the most important features as those whose
importance values are greater than 1.5 times the standard deviation of the
importance values, up to a maximum of 100 features.

##  3 Experiments and results

###  3.1 Dataset and black-box model

The Commercial Modular Aero-Propulsion System Simulation (CMAPSS) is a
modeling software developed at NASA. It was used to build the well known
CMAPSS dataset [Saxena(2008)] as well as the recently created N-CMAPSS dataset
[Arias(2021)]. N-CMAPSS was created providing the full history of the
trajectories starting with a healthy condition until the failure occurs. A
schematic of the turbofan model used in the simulations is shown in Figure 4.
All rotation components of the engine (fan, LPC, HPC, LPT, and HPT) can be
affected by the degradation process.

Figure 4: Schematic of the model used in N-CMAPSS [Arias(2021)].

Seven different failure modes, related to flow degradation or subcomponent
efficiency, that can be present in each flight have been defined. The flights
are divided into three classes depending on the length of the flight. Flights
with a duration from 1 to 3 hours belong to class 1, class 2 consists of
flights between 3 and 5 hours, and flights that take more than 5 hours fall
into class 3. Each flight is divided into cycles, covering climb, cruise, and
descend operations.

The problem revolves around the development of a model $h$ capable of
predicting the remaining useful life $y$ of the system, using the sensor
outputs $x_{s}$, the scenario descriptors $w$ and auxiliary data $a$. The
different variables available to estimate the RUL of the system are described
in table 1. The former is an optimization problem that can be denoted as:

$\underset{h\>\in\>\mathcal{H}}{argmin}\>\sum\mathcal{S}\left(y-h(x_{s},w,a)\right)$
(22)

where $y$ and $h(x_{s},w,a)$ are, respectively, the expected and estimated
RUL. $\mathcal{H}$ is the set of the different models to be tested by the
optimization process and $\mathcal{S}$ is a scoring function defined as the
average of the Root-Mean-Square Error (RMSE) and the NASA’s scoring function
($N_{s}$) [Saxena(2008)]:

$\mathcal{S}=0.5\cdot RMSE+O.5\cdot N_{s}$ (23)

$N_{s}=\frac{1}{M}\sum exp(\alpha|y-\hat{y}|)-1$ (24)

being $M$ the number of samples and being $\alpha$ equal to $\frac{1}{13}$ in
case that $\hat{Y}<Y$ and $\frac{1}{10}$ otherwise.

Table 1: Variable description, symbol, units and variable set. Symbol | Set | Description | Units
---|---|---|---
alt | $W$ | Altitude | ft
Mach | $W$ | Flight Mach number | -
TRA | $W$ | Throttle-resolver angle | %
T2 | $W$ | Total temperature at fan inlet | ºR
Wf | $X_{s}$ | Fuel flow | pps
Nf | $X_{s}$ | Physical fan speed | rpm
Nc | $X_{s}$ | Physical core speed | rpm
T24 | $X_{s}$ | Total temperature at LPC outlet | ºR
T30 | $X_{s}$ | Total temperature at HPC outlet | ºR
T48 | $X_{s}$ | Total temperature at HPT outlet | ºR
T50 | $X_{s}$ | Total temperature at LPT outlet | ºR
P15 | $X_{s}$ | Total pressure in bypass-duct | psia
P2 | $X_{s}$ | Total pressure at fan inlet | psia
P21 | $X_{s}$ | Total pressure at fan outlet | psia
P24 | $X_{s}$ | Total pressure at LPC outlet | psia
Ps30 | $X_{s}$ | Static pressure at HPC outlet | psia
P40 | $X_{s}$ | Total pressure at burner outlet | psia
P50 | $X_{s}$ | Total pressure at LPT outlet | psia
Fc | $A$ | Flight class | -
$h_{s}$ | $A$ | Health state | -

The model used in the experimentation was designed and implemented by the
authors and it achieved the third place in the 2021 PHM Conference Data
Challenge [Solis(2021].

The former 20 variables have different scales, thus a z-score normalization is
applied to homogenize the variables scale:

$x^{\prime}_{f}=\frac{x_{f}-\mu_{f}}{\sigma_{f}}$ (25)

where $x_{f}$ is the data of a feature $f$, and $\mu_{f}$ and $\sigma_{f}$ are
its mean and standard deviation, respectively.

The network inputs are generated by sliding a time window through the
normalized data, with the window size denoted as $L_{w}$ and determined during
model selection. The inputs are defined as

$X^{k}_{t}=[\widetilde{\mathcal{X}}^{k}_{t_{end}-L_{w}},...,\widetilde{\mathcal{X}}^{k}_{t_{end}}]$

where $t_{end}$ is the end time of the window (see Figure 5). The
corresponding ground truth RUL label for each input is denoted as $Y_{t}$.
This method generates $T^{k}-L_{w}$ samples for each unit, where $T^{k}$
represents the total run time in seconds of the unit.

Figure 5: Sliding window.

The ground-RUL label has been defined as a linear function of cycles from the
RUL of each unit $Y^{k}_{t}=TUL^{k}-C^{k}_{t}$, where $TUL^{k}$ is the total
useful life of the unit $k$ in cycles and $C^{k}_{t}$ is the number of past
cycles from the beginning of the experiment at time $t$.

The black-box model is a Deep Convolutional Neural Network (DCNN). The
classical DCNN architecture, which is shown in Figure 6, can be divided into
two parts. The architecture is composed of two main parts. The first part
consists of a stacking of $N_{b}$ blocks that include convolutional and
pooling layers. The main objective of this part is to extract relevant
features for the task at hand. The second part is made up of fully connected
layers, which are responsible for performing the regression of the RUL in this
case. The detailed parameters of the network can be found in Table 2. The
network is composed of four convolutional blocks, each of which consists of
four stacked convolutional layers with hyperbolic tangent ($\tanh$) activation
functions. Following the convolutional blocks, two fully connected layers with
leaky ReLU activation are stacked. The output layer uses the rectified linear
unit (ReLU) activation function, since negative values in the output are not
desired.

Algorithm 1 Algorithm to compute each proxy on the test set

$X$ is a set of $N$ samples

$P$ is a proxy

$N\leftarrow\left|X\right|$

$S\leftarrow 0$

for $x_{i}\in X$ do

$s_{i}\leftarrow P(x_{i})$

$S\leftarrow S+s_{i}$

end for

return $\frac{S}{N}$

Figure 6: DCNN network architecture used in experiments. Table 2: Parameters of the black-box model developed Patameter | Value
---|---
$L_{w}$ | 161
$B_{s}$ | 116
$C_{bs}$ | 4
$N_{cb}$ | 4
$fc_{1}$ | 256
$fc_{2}$ | 100
$\sigma_{conv}$ | tanh
$d_{rate}$ | 2
$\sigma_{fc}$ | Leaky ReLU
$\sigma_{output}$ | ReLU
#Net params | 1,514,016
RMSE | 10.46
MAE | 7.689
NASA score | 2.13
CV $\mathcal{S}$ score | 6.30
std($\mathcal{S}$) | 0.37

###  3.2 Experiments

The experiments were conducted using a dataset consisting of 256 data samples.
These samples were not used during the model training phase. Each data sample
contains 20 time series, and each time series consists of 160 time units. The
8 proxies, that were defined in Section 2.3, were computed for each sample,
and the final score for each proxy was determined by taking the mean of the
256 samples as it is described in the algorithm 1. This process was repeated
for each XAI method studied.

For both LIME and SHAP, five perturbation methods were tested: zero, one,
mean, uniform noise, and normal noise. In the case of selectivity, due to
performance issues, groups of 10 features, ordered by importance, were
considered for computing the AUC. For the remaining XAI methods, the replace-
by-zero perturbation method has been used since, in the experiments performed,
it provided the best results. Finally, in Grad-CAM, different values between 0
and 1 for $\beta$ and $\sigma$ (factors adjusting the contribution of the time
and feature components in the computation of feature importance) were tested
using a grid search methodology with a step of 0.1. Furthermore, heat maps
were extracted for each convolutional layer to study the best layer for
explaining the predictions of the DCNN model on this dataset. It is important
to note that the gradients are distributed differently depending on its depth
within the network (Figure 7), which means that the last convolutional layer,
commonly exploited in the literature, may not be the best to solve all
problems. By using the proxies, it is possible to assess which is the best
layer from the perspective of explainability.

Figure 7: Grad-CAM feature map distribution over the input depending on its
depth within the network

###  3.3 Results

The results for each method are presented in Table 3. This table shows that
Grad-CAM achieved the highest value for 5 out of the 8 proxies tested, and for
selectivity and completeness, the value was close to the maximum. The best
result for Grad-CAM was obtained using $\beta=0.9$, $\sigma=0.0$, and
computing the gradient with respect to the second layer of the network. Note
that Grad-CAM gets the worst result, compared to the rest of the proxies,
according to the acumen proxy, defined in this work.

Table 3: This table shows the result of the models for the different proxies. Perm: Permutation, I: Identity, Sep: Separability, Sel: Selectivity, Sta: Stability, Coh: Coherence, Comp: Completeness, Cong: Congruency, Acu: Acumen Method | Perm | I | Sep | Sta | Sel | Coh | Comp | Cong | Acu
---|---|---|---|---|---|---|---|---|---
Saliency |  | 1.000 | 0.999 | 0.055 | 0.450 | 0.174 | 0.972 | 0.163 | 0.516
LRP |  | 1.000 | 1.000 | -0.037 | 0.599 | 0.180 | 0.967 | 0.165 | 0.495
Lime | mean | 0.004 | 1.000 | 0.130 | 0.573 | 0.173 | 0.962 | 0.161 | 0.685
Lime | n. noise | 0.008 | 1.000 | 0.131 | 0.582 | 0.173 | 0.960 | 0.162 | 0.677
Lime | u. noise | 0.012 | 1.000 | 0.109 | 0.560 | 0.166 | 0.960 | 0.162 | 0.577
Lime | zero | 1.000 | 1.000 | 0.554 | 0.835 | 0.16 | 1.017 | 0.146 | 0.753
Lime | one | 1.000 | 1.000 | 0.349 | 0.728 | 0.184 | 0.969 | 0.166 | 0.069
Grad-CAM |  | 1.000 | 1.000 | 0.653 | 0.741 | 0.196 | 0.947 | 0.171 | 0.435
SHAP | mean | 0.000 | 1.000 | 0.033 | 0.582 | 0.120 | 0.973 | 0.152 | 0.505
SHAP | n. noise | 0.000 | 1.000 | 0.037 | 0.581 | 0.116 | 0.968 | 0.150 | 0.501
SHAP | u. noise | 0.000 | 1.000 | 0.027 | 0.581 | 0.125 | 0.961 | 0.162 | 0.503
SHAP | zero | 1.000 | 1.000 | 0.226 | 0.800 | 0.152 | 1.001 | 0.149 | 0.761
SHAP | one | 1.000 | 1.000 | 0.200 | 0.692 | 0.173 | 0.969 | 0.169 | 0.349

Since Grad-CAM is clearly the method providing the best results, further
analysis have been carried out to understand the behavior of the method, under
different settings, for each of the proxies studied. The identity proxy was
omitted, as it is always 1 in Grad-CAM. Figure 8 compares the scores obtained
by Grad-CAM when applied to each convolutional layer of the DCNN. The
selectivity, stability, separability, and coherence proxies show an inverse
correlation with respect to the depth of the layer. Conversely, the proxies
for completeness, congruency and acumen present direct correlation with the
layer depth.

The inverse correlation may be due to the existence of large groups of
features having a higher likelihood of including features that impact the
proxy negatively. For example, in the case of selectivity, a group that is
considered important as a whole could contain samples with low importance.
Therefore, for these proxies it is better to consider features independently,
instead of as a group. Figure 9 shows the influence of the time contribution,
which is controlled by the factor $\beta$. The factor being discussed shows a
direct correlation with all proxies excepting congruency and coherence, which
present an inverse correlation. The reason for this effect on coherence may be
due to the introduction of the time contribution, which spreads the importance
across a larger number of features. As a consequence of this, the limit in
considering a maximum of 100 features impacts negatively on these two proxies.
Lastly, the feature contribution, weighted by the factor $\sigma$, was found
to have no contribution in improving any of the proxy scores.

Figure 8: Grad-CAM behavior for each proxy in each layer Figure 9: Grad-CAM
behavior for each proxy respect $\beta$

##  4 Discussion

This work is focused on the under-researched area of XAI methods for time
series and regression problems. The first aim of this work was to review
existing papers on XAI addressing such topics, with an emphasis in the use of
quantitative metrics to compare XAI methods. Then, a comparison among the most
promising XAI methods was carried out on a highly complex model, as is the
DCNN, applied to a time series regression problem within the context of PHM.
With this aim, a number of experiments were performed, quantifying the quality
of explanations by computing eight different proxies. Results showed that
Grad-CAM was the most effective XAI method among the ones tested, achieving
the highest values for five out of eight proxies and being close to the
maximum in two others. The results also showed that the impact of the layers
and the time component contribution $\beta$ on Grad-CAM varied for different
proxies, showing some of them a direct correlation and others an inverse
correlation. The findings of this study highlight the need for further
research in this area and the importance of developing better XAI methods for
time series and regression problems, particularly in the PHM field.

In addition to comparing various XAI methods through quantitative proxies,
this work also makes a significant contribution by introducing a new
quantitative proxy called acumen, which measures a desirable property of any
XAI method and highlights the breach of this property by Grad-CAM.
Furthermore, an extension of Grad-CAM that takes into account time
dependencies is proposed (where such contribution can be modulated), and
results showed that this extension improves the performance of most of the
studied XAI methods. These findings demonstrate the importance of considering
time dependencies when evaluating the performance of XAI methods in time
series and provide valuable insights for future research in this area.

The experiments were carried out on a deep neural network trained to predict
the remaining useful life of a turbine engine, which belongs to a type of
regression problem that has received little attention in XAI. All the code to
reproduce the experiments, along with the data and the model, is provided to
allow the research community to further explore these findings. Overall, this
work makes a valuable contribution to the field of XAI by addressing important
gaps in the literature and presenting novel approaches for time series and
regression problems.

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