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Vera Serganova

Publications and source records attributed to Vera Serganova.

At least 37 records · Page 2Linked to original sources
arXiv2023

Splitting quasireductive supergroups and volumes of supergrassmannians

We introduce the notion of splitting subgroups of quasireducitve supergroups, and explain their significance. For $GL(m|n)$, $Q(n)$, and defect one basic classical supergroups, we give explicit splitting subgroups. We further prove they are minimal up to conjugacy, except in the $GL(m|n)$ case where it remains a conjecture. A key tool in the proof is the computation of the volumes of complex supergrassmannians, which is of interest in its own right.

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arXiv2023

It takes two spectral sequences

We study the representation theory of the Lie superalgebra $\mathfrak{gl}(1|1)$, constructing two spectral sequences which eventually annihilate precisely the superdimension zero indecomposable modules in the finite-dimensional category. The pages of these spectral sequences, along with their limits, define symmetric monoidal functors on $\mathrm{Rep} (\mathfrak{gl}(1|1))$. These two spectral sequences are related by contragredient duality, and from their limits we construct explicit semisimplification functors, which we explicitly prove are isomorphic up to a twist. We use these tools to prove branching results for the restriction of simple modules over Kac-Moody and queer Lie superalgebras to $\mathfrak{gl}(1|1)$-subalgebras.

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arXiv2022

Deligne categories and the limit of categories $Rep(GL(m|n))$

For each integer $t$ a tensor category $V_t$ is constructed, such that exact tensor functors $V_t \longrightarrow C$ classify dualizable $t$-dimensional objects in $C$ not annihilated by any Schur functor. This means that $V_t$ is the "abelian envelope" of the Deligne category $Rep(GL_t)$. Any tensor functor $Rep(GL_t)\longrightarrow C$ is proved to factor either through $V_t$ or through one of the classical categories $Rep(GL(m|n))$ with $m-n=t$. The universal property of $V_t$ implies that it is equivalent to the categories $Rep_{Rep(GL_{t_1})\otimes Rep(GL_{t_2})}(GL(X),ε)$, ($t=t_1+t_2$, $t_1$ not integer) suggested by Deligne as candidates for the role of abelian envelope.

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arXiv2022

Bounded weight modules for basic classical Lie superalgebras at infinity

We classify simple bounded weight modules over the complex simple Lie superalgebras $\mathfrak{sl}(\infty |\infty)$ and $\mathfrak{osp} (m | 2n)$, when at least one of $m$ and $n$ equals $\infty$. For $\mathfrak{osp} (m | 2n)$ such modules are of spinor-oscillator type, i.e., they combine into one the known classes of spinor $\mathfrak{o} (m)$-modules and oscillator-type $\mathfrak{sp} (2n)$-modules. In addition, we characterize the category of bounded weight modules over $\mathfrak{osp} (m | 2n)$ (under the assumption $\dim \, \mathfrak{osp} (m | 2n) = \infty$) by reducing its study to already known categories of representations of $\mathfrak{sp} (2n)$, where $n$ possibly equals $\infty$. When classifying simple bounded weight $\mathfrak{sl}(\infty |\infty)$-modules, we prove that every such module is integrable over one of the two infinite-dimensional ideals of the Lie algebra $\mathfrak{sl}(\infty |\infty)_{\bar{0}}$. We finish the paper by establishing some first facts about the category of bounded weight $\mathfrak{sl} (\infty |\infty)$-modules.

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arXiv2021

Capelli operators for spherical superharmonics and the Dougall-Ramanujan identity

Let $(V,ω)$ be an orthosympectic $\mathbb Z_2$-graded vector space and let $\mathfrak g:=\mathfrak{gosp}(V,ω)$ denote the Lie superalgebra of similitudes of $(V,ω)$. When the space $\mathscr P(V)$ of superpolynomials on $V$ is \emph{not} a completely reducible $\mathfrak g$-module, we construct a natural basis $D_λ$ of Capelli operators for the algebra of $\mathfrak g$-invariant superpolynomial superdifferential operators on $V$, where the index set $\mathcal P$ is the set of integer partitions of length at most two. We compute the action of the operators $D_λ$ on maximal indecomposable components of $\mathscr P(V)$ explicitly, in terms of Knop-Sahi interpolation polynomials. Our results show that, unlike the cases where $\mathscr P(V)$ is completely reducible, the eigenvalues of a subfamily of the $D_λ$ are \emph{not} given by specializing the Knop-Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. In addition, we demonstrate a close relationship between our eigenvalue formulas for this subfamily of Capelli operators and the Dougall-Ramanujan hypergeometric identity. We also transcend our results on the eigenvalues of Capelli operators to the Deligne category $\mathsf{Rep}(O_t)$. More precisely, we define categorical Capelli operators $\{\mathbf D_{t,λ}\}_{λ\in\mathcal P}^{}$ that induce morphisms of indecomposable components of symmetric powers of $\mathsf V_t$, where $\mathsf V_t$ is the generating object of $\mathsf{Rep}(O_t)$. We obtain formulas for the eigenvalue polynomials associated to the $\left\{\mathbf D_{t,λ}\right\}_{λ\in\mathcal P}$ that are analogous to our results for the operators $\{D_λ\}_{λ\in\mathcal P}^{}$.

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arXiv2020

Extension Quiver for Lie Superalgebra $\mathfrak{q}(3)$

We describe all blocks of the category of finite-dimensional $\mathfrak{q}(3)$-supermodules by providing their extension quivers. We also obtain two general results about the representation of $\mathfrak{q}(n)$: we show that the Ext quiver of the standard block of $\mathfrak{q}(n)$ is obtained from the principal block of $\mathfrak{q}(n-1)$ by identifying certain vertices of the quiver and prove a ''virtual'' BGG-reciprocity for $\mathfrak{q}(n)$. The latter result is used to compute the radical filtrations of $\mathfrak{q}(3)$ projective covers.

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arXiv2020

Gelfand-Tsetlin modules for gl(n,m)

We address the problem of classifying of irreducible Gelfand-Tsetlin modules for gl(m|n) and show that it reduces to the classification of Gelfand-Tsetlin modules for the even part. We also give an explicit tableaux construction and the irreducibility criterion for the class of quasi typical and quasi covariant Gelfand-Tsetlin modules which includes all essentially typical and covariant tensor finite dimensional modules. In the quasi typical case new irreducible representations are infinite dimensional gl(m|n)-modules which are isomorphic to the parabolically induced (Kac) modules.

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arXiv2019

Deligne categories and the periplectic Lie superalgebra

We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $\mathfrak{p}(n)$ as $n \to \infty$. The paper gives a construction of the tensor category $Rep(\underline{P})$, possessing nice universal properties among tensor categories over the category $\mathtt{sVect}$ of finite-dimensional complex vector superspaces. First, it is the "abelian envelope" of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699. Secondly, given a tensor category $\mathcal{C}$ over $\mathtt{sVect}$, exact tensor functors $Rep(\underline{P})\longrightarrow \mathcal{C}$ classify pairs $(X, ω)$ in $\mathcal{C}$ where $ω: X \otimes X \to Π\mathbf{1}$ is a non-degenerate symmetric form and $X$ not annihilated by any Schur functor. The category $Rep(\underline{P})$ is constructed in two ways. The first construction is through an explicit limit of the tensor categories $Rep(\mathfrak{p}(n))$ ($n\geq 1$) under Duflo-Serganova functors. The second construction (inspired by P. Etingof) describes $Rep(\underline{P})$ as the category of representations of a periplectic Lie supergroup in the Deligne category $\mathtt{sVect} \boxtimes Rep(\underline{GL}_t)$. An upcoming paper by the authors will give results on the abelian and tensor structure of $Rep(\underline{P})$.

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arXiv2019

Duflo-Serganova functor and superdimension formula for the periplectic Lie superalgebra

In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo-Serganova functor. Given a simple $\mathfrak{p}(n)$-module $L$ and a certain element $x\in \mathfrak{p}(n)$ of rank $1$, we give an explicit description of the composition factors of the $\mathfrak{p}(n-1)$-module $DS_x(L)$, which is defined as the homology of the complex $$ΠM \xrightarrow{x} M \xrightarrow{x} ΠM.$$ In particular, we show that this $\mathfrak{p}(n-1)$-module is multiplicity-free. We then use this result to give a simple explicit combinatorial formula for the superdimension of a simple integrable finite-dimensional $\mathfrak{p}(n)$-module, based on its highest weight. In particular, this reproves the Kac-Wakimoto conjecture for $\mathfrak{p}(n)$, which was proved earlier by the authors.

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arXiv2019

Translation functors and decomposition numbers for the periplectic Lie superalgebra $\mathfrak{p}(n)$

We study the category $\mathcal{F}_n$ of finite-dimensional integrable representations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. We define an action of the Temperley--Lieb algebra with infinitely many generators and defining parameter $0$ on the category $\mathcal{F}_n$ by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m|n)$. Using the Temperley--Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of $\mathcal{F}_n$. We also prove that indecomposable projective modules in this category are multiplicity-free.

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arXiv2019

Tensor product of the Fock representation with its dual and the Deligne category

We describe the structure of the tensor product of the basic Fock representation of sl(\infty) with its shifted dual. More precisely we prove that this tensor product has a unique decreasing filtration with simple quotients. We use the categorification of this representation via translation functors in the abelain envelope of the Deligne category GL(t) for integral t. We also compute dimensions of standard and tilting objects in this abelian envelope.

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arXiv2019

Snowflake modules and Enright functor for Kac-Moody superalgebras

We introduce a class of modules over Kac-Moody superalgebras; we call these modules snowflake. These modules are characterized by invariance property of their characters with respect to a certain subgroup of the Weyl group. Examples of snowflake modules appear as admissible modules in representation theory of affine vertex algebras and in classification of bounded weight modules. Using these modules we prove Arakawa's Theorem for the Lie superalgebra osp(1|2n).

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arXiv2019

Grothendieck rings of periplectic Lie superalgebras

We describe explicitly the Grothendieck rings of finite-dimensional representations of the periplectic Lie superalgebras. In particular, the Grothendieck ring of the Lie supergroup $P(n)$ is isomorphic to the ring of symmetric polynomials in $x_1^{\pm 1}, \ldots, x_n^{\pm 1}$ whose evaluation $x_1=x_2^{-1}=t$ is independent of $t$.

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