Kac-Wakimoto conjecture for the periplectic Lie superalgebra

TL;DR

This paper proves the Kac-Wakimoto conjecture for the periplectic Lie superalgebra, establishing that simple modules in non-maximal atypicality blocks have zero superdimension, advancing understanding of superalgebra representation theory.

Contribution

The paper provides a proof of the Kac-Wakimoto conjecture specifically for the periplectic Lie superalgebra, a case previously unresolved.

Findings

Simple modules in non-maximal atypicality blocks have superdimension zero

The conjecture holds for the periplectic Lie superalgebra

Advances the classification of modules in superalgebra theory

Abstract

We prove the Kac-Wakimoto conjecture for the periplectic Lie superalgebra $\mathfrak{p}(n)$, stating that any simple module lying in a block of non-maximal atypicality has superdimension zero.

Equations

$${\Theta }_k(M^{\ast })\cong \Pi {\left({\Theta }_{-k}(M)\right)}^{\ast }$$

$$coev:\text{mathbbm}1\to V\otimes V^{\ast },ev:V^{\ast }\otimes V\to \text{mathbbm}1$$

$$\beta :V_n\otimes V_n\to \mathbb{C},\beta (v,w)=\beta (w,v),\text{and}\beta (v,w)=0\text{if}p(v)=p(w).$$

$$\beta (Xv,w)+(-1{)}^{p(X)p(v)}\beta (v,Xw)=0.$$

$${\mathfrak{p}(n)}_0\cong \mathfrak{gl}(n),{\mathfrak{p}(n)}_{-1}\cong \Pi {\wedge }^2({\mathbb{C}}^n{)}^{\ast },{\mathfrak{p}(n)}_1\cong \Pi S^2{\mathbb{C}}^n.$$

$${\wedge }^2{V}_n\twoheadrightarrow L_n(-2{\epsilon }_1),L_n(-{\epsilon }_1-{\epsilon }_2)\hookrightarrow S^2{V}_n.$$

$$\bar{\lambda }=\lambda +{\rho }^{(n)}.$$

$$\mathfrak{gl}(V_n)\cong {\mathfrak{p}}_n\oplus {\mathfrak{p}}_n^{\ast }$$

$$tr:=ev\circ {\sigma }_{V_n,V_n^{\ast }}:\mathfrak{gl}(V_n)\cong V_n\otimes V_n^{\ast }\to \mathbb{C}.$$

$$V_n\otimes M\stackrel{\mathrm{Id}\otimes coev\otimes \mathrm{Id}}{\to }{V}_n\otimes \mathfrak{p}(n{)}^{\ast }\otimes \mathfrak{p}(n)\otimes M\stackrel{i_{\ast }\otimes \mathrm{Id}}{\to }{V}_n\otimes \mathfrak{gl}(V_n)\otimes \mathfrak{p}(n)\otimes M\stackrel{act\otimes act}{\to }{V}_n\otimes M$$

$${{\Theta }^{\prime }}_k^{(n)}(M):=\bigcup _{m>0}\mathrm{Ker}({\Omega }^{(n)}-k\mathrm{Id}{)}_{{|}_{M\otimes V_n}}^m$$

$${\mathcal{F}}_n=\underset{k\in \{-n,-n+2,\dots ,n-2,n\}}{⨁}{\left({\mathcal{F}}_n\right)}_k^{+}\oplus \underset{k\in \{-n,-n+2,\dots ,n-2,n\}}{⨁}{\left({\mathcal{F}}_n\right)}_k^{-},$$

$${\mathcal{F}}_n^k:={\left({\mathcal{F}}_n\right)}_k^{+}\oplus {\left({\mathcal{F}}_n\right)}_k^{-}.$$

$${\Theta }_i{\mathcal{F}}_n^k\subset \{\begin{array}{l}{\mathcal{F}}_n^{k+2}\text{if}i\text{is odd}\\ {\mathcal{F}}_n^{k-2}\text{if}i\text{is even}\end{array}$$

$${\Theta }_I^{(n)}:={\Theta }_{i_1}^{(n)}\circ {\Theta }_{i_2}^{(n)}\circ \dots \circ {\Theta }_{i_k}^{(n)}$$

$$t(I):=\sum _{s=0}^k(-1{)}^{i_s+1}.$$

$$M\in {\mathcal{F}}_l^n⟹{\Theta }_I^{(n)}M\in {\mathcal{F}}_{l+2t(I)}^n.$$

$${DS}_{x}(M)=\raise 4.30554pt\hbox{$Ker(x\rvert_{M})$}\Big{/}\lower 4.30554pt\hbox{$Im(x\rvert_{M})$}.$$

$$D{S}_x{\Theta }_k^{(n)}\stackrel{\sim }{⟶}{\Theta }_k^{(n-s)}D{S}_x$$

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\&&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{DS_{x}\Theta^{(n)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 75.34554pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\eta}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 90.96048pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 15.10751pt\raise-20.06943pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{DS_{x}\left(\Omega_{n}\right)}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 46.98024pt\raise-28.20554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 90.96048pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Theta^{(n-s)}DS_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 113.18655pt\raise-20.06943pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.9611pt\hbox{$\scriptstyle{\Omega_{n-s}DS_{x}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 113.18655pt\raise-28.20554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-40.13885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise-40.13885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{DS_{x}\Theta^{(n)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 75.34554pt\raise-34.95135pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\eta}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 90.96048pt\raise-40.13885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 90.96048pt\raise-40.13885pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Theta^{(n-s)}DS_{x}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$$

$${\Theta }_I^{(n)}L\in {\mathcal{F}}_n^n\text{ and }{\Theta }_I^{(n)}L\ne 0.$$

$${\Theta }_I^{(n)}L\in {\mathcal{F}}_{-n}^n\text{ and }{\Theta }_I^{(n)}L\ne 0.$$

$${\Theta }_I^{(n-2)}:={\Theta }_{i_1}^{(n-2)}\circ {\Theta }_{i_2}^{(n-2)}\circ \dots \circ {\Theta }_{i_s}^{(n-2)}$$

$${\Theta }_I^{(n-2)}D{S}_x{L}^{\prime }\ne 0\text{ and }D{S}_x{L}^{\prime }\in {\mathcal{F}}_{n-2}^{n-2}.$$

$${\Theta }_I^{(n)}:={\Theta }_{i_1}^{(n)}\circ {\Theta }_{i_2}^{(n)}\circ \dots \circ {\Theta }_{i_s}^{(n)}$$

$${\Theta }_I^{(n-2)}D{S}_{x}L\cong D{S}_x{\Theta }_I^{(n)}L.$$

$${\Theta }_I^{(n)}L={\Theta }_{i_1}^{(n)}\circ {\Theta }_{i_2}^{(n)}\circ \dots \circ {\Theta }_{i_s}^{(n)}L\in {\mathcal{F}}_n^{k+2t(I)}$$

$${\mu }_k+k\equiv {\mu }_n-n(\mathrm{mod}2)$$

$${\mathrm{Hom}}_{\mathfrak{p}(n)}(P^{\prime },{\Theta }_I^{(n)}L)={\mathrm{Hom}}_{\mathfrak{p}(n)}({\Theta }_J^{(n)}{P}^{\prime },L)={\mathrm{Hom}}_{\mathfrak{p}(n)}(P,L)=\mathbb{C}.$$

Full text

1. Introduction

1.1.

Consider a complex vector superspace $V$, and let ${\mathbb{C}}^{0|1}$ be the odd one-dimensional vectors superspace.

The (complex) periplectic Lie superalgebra $\mathfrak{p}(V)$ is the Lie superalgebra of endomorphisms of a complex vector superspace $V$ possessing a non-degenerate symmetric form $\omega:{S^{2}V}\to{\mathbb{C}}^{0|1}$ (this form is also referred to as an “odd form”). An example of such superalgebra is $\mathfrak{p}(n)=\mathfrak{p}({\mathbb{C}}^{n|n})$ for $V={\mathbb{C}}^{n|n}$, where $\omega_{n}:{\mathbb{C}}^{n|n}\otimes{\mathbb{C}}^{n|n}\to{\mathbb{C}}^{0|1}$ pairing the even and odd parts of the vector superspace ${\mathbb{C}}^{n|n}$.

The periplectic Lie superalgebras has an interesting non-semisimple representation theory; some results on the category ${\mathcal{F}}_{n}$ of finite-dimensional integrable representations of $\mathfrak{p}(n)$ can be found in [BDE*+*16, Che15, Cou16, DLZ15, Gor01, Moo03, Ser02].

In [BDE*+*16], the blocks of the category ${\mathcal{F}}_{n}$ were classified: it was shown that (up to change of pairity) the blocks can be numbered by integers $-n,-n+2,-n+4,\ldots,n-4,n-2,n$, with the trivial representation sitting in block number $n(\mathrm{mod}2)$. We denote this blocks by ${\mathcal{F}}^{k}_{n}$, $k\in\{-n,-n+2,\ldots,n-2,n\}$.

In this article, we prove the following version of the Kac-Wakimoto conjecture:

Theorem 1**.**

We have ${\operatorname{sdim}}M=0$ if $M\in{\mathcal{F}}^{k}_{n}$, $k\neq 0,\pm 1$, where ${\operatorname{sdim}}$ denotes the superdimension (${\operatorname{sdim}}M=\dim M_{\bar{0}}-\dim M_{\bar{1}}$).

The main ingredients in the proof of this theorem, are the translation functors acting on ${\mathcal{F}}_{n}$, and the Duflo-Serganova functor $DS:{\mathcal{F}}_{n}\to{\mathcal{F}}_{n-2}$. The translation functors are direct summands of the functor $-\otimes V$, whose action on the blocks ${\mathcal{F}}_{n}$ was obtained in [BDE*+*16]; the functor $DS:{\mathcal{F}}_{n}\to{\mathcal{F}}_{n-2}$ is a tensor functor preserving dimension, which allows us to reduce the problem of computing dimensions in ${\mathcal{F}}_{n}$ to a similar problem in ${\mathcal{F}}_{n-2}$.

We also prove the following statement:

Theorem 2**.**

Let $M\in{\mathcal{F}}_{n}$ be an object lying in a certain block as described in Section 2.3.5 and [BDE*+*16].

1. (1)

The object $M^{*}$ also lies in the same block of ${\mathcal{F}}_{n}$. 2. (2)

We have a natural isomorphism

[TABLE]

where $\Theta_{k}$ is the $k$-th translation functor on ${\mathcal{F}}_{n}$ (see Definition 2.3.5).

1.2. Acknowledgements

I.E.-A. was supported by the ISF grant no. 711/18. V.S. was supported by NSF grant 1701532.

2. Preliminaries

2.1. General

Throughout this paper, we will work over the base field $\mathbb{C}$, and all the categories considered will be ${\mathbb{C}}$-linear.

A vector superspace will be defined as a $\mathbb{Z}/2\mathbb{Z}$-graded vector space $V=V_{\bar{0}}\oplus V_{\bar{1}}$. The parity of a homogeneous vector $v\in V$ will be denoted by $p(v)\in\mathbb{Z}/2\mathbb{Z}=\{\bar{0},\bar{1}\}$ (whenever the notation $p(v)$ appears in formulas, we always assume that $v$ is homogeneous).

2.2. Tensor categories

In the context of symmetric monoidal (SM) categories, we will denote by ${\mathbbm{1}}$ the unit object, and by $\sigma$ the symmetry morphisms.

A functor between symmetric monoidal categories will be called a SM functor if it respects the SM structure.

Given an object $V$ in a SM category, we will denote by

[TABLE]

the coevaluation and evaluation maps for $V$. We will also denote by $\mathfrak{gl}(V):=V\otimes V^{*}$ the internal endomorphism space with the obvious Lie algebra structure on it. The object $V$ is then a module over the Lie algebra $\mathfrak{gl}(V)$; we denote the action by $act:\mathfrak{gl}(V)\otimes V\to V$. For two functors $F,G$ we write $F\vdash G$ if $F$ is left adjoint of $G$.

2.3. The periplectic Lie superalgebra

2.3.1. Definition of the periplectic Lie superalgebra

Let $n\in{\mathbb{Z}}_{>0}$, and let $V_{n}$ be an $(n|n)$-dimensional vector superspace equipped with a non-degenerate odd symmetric form

[TABLE]

Then $\operatorname{End}_{\mathbb{C}}(V_{n})$ inherits the structure of a vector superspace from $V_{n}$. We denote by $\mathfrak{p}(n)$ the Lie superalgebra of all $X\in\operatorname{End}_{\mathbb{C}}(V_{n})$ preserving $\beta$, i.e. satisfying

[TABLE]

Remark 2.3.1*.*

Choosing dual bases $v_{1},v_{2},\ldots,v_{n}$ in $V_{\bar{0},n}$ and $v_{1^{\prime}},v_{2^{\prime}},\ldots v_{n^{\prime}}$ in $V_{\bar{1},n}$, we can write the matrix of $X\in\mathfrak{p}(n)$ as $\left(\begin{smallmatrix}A&B\\ C&-A^{t}\end{smallmatrix}\right)$ where $A,B,C$ are $n\times n$ matrices such that $B^{t}=B,\,C^{t}=-C$.

We will also use the triangular decomposition ${\mathfrak{p}(n)}\cong{\mathfrak{p}(n)}_{-1}\oplus{\mathfrak{p}(n)}_{0}\oplus{\mathfrak{p}(n)}_{1}$ where

[TABLE]

Then the action of $\mathfrak{p}(n)_{\pm 1}$ on any $\mathfrak{p}(n)$-module is ${\mathfrak{p}(n)}_{0}$-equivariant.

2.3.2. Weights for the periplectic

superalgebra

The integral weight lattice for $\mathfrak{p}(n)$ will be $span_{\mathbb{Z}}\{{\varepsilon}_{i}\}_{i=1}^{n}$.

• $\star$

We fix a set of simple roots ${\varepsilon}_{2}-{\varepsilon}_{1},\ldots,{\varepsilon}_{n}-{\varepsilon}_{n-1},{-{\varepsilon}_{n-1}-{\varepsilon}_{n}}$, the last root is odd and all others are even.

Hence the dominant integral weights will be given by $\lambda=\sum_{i}\lambda_{i}{\varepsilon}_{i}$, where $\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{n}$.

• $\star$

We fix an order on the weights of $\mathfrak{p}(n)$: for weights $\mu,\lambda$, we say that $\mu\geq\lambda$ if $\mu_{i}\leq\lambda_{i}$ for each $i$.

Remark 2.3.2*.*

It was shown in [BDE*+*16, Section 3.3] that if $\leq$ corresponds to a highest-weight structure on the category of finite-dimensional representations of $\mathfrak{p}(n)$. Note that in the cited paper we use slightly different set of simple roots $-{\varepsilon}_{1}-{\varepsilon}_{2},{\varepsilon}_{1}-{\varepsilon}_{2},\ldots,{\varepsilon}_{n-1}-{\varepsilon}_{n}$.

• $\star$

The simple finite-dimensional representation of $\mathfrak{p}(n)$ corresponding to the weight $\lambda$ whose highest weight vector is even will be denoted by $L_{n}(\lambda)$.

Example 2.3.3**.**

Let $n\geq 2$. The natural representation $V_{n}$ of $\mathfrak{p}(n)$ has highest weight $-{\varepsilon}_{1}$, with odd highest-weight vector; hence $V_{n}\cong\Pi L_{n}(-{\varepsilon}_{1})$. The representation $\bigwedge^{2}V_{n}$ has highest weight $-2{\varepsilon}_{1}$, and the representation $S^{2}V_{n}$ has highest weight $-{\varepsilon}_{1}-{\varepsilon}_{2}$; both have even highest weight vectors, so

[TABLE]

• $\star$

Set $\rho^{(n)}=\sum_{i=1}^{n}(i-1){\varepsilon}_{i}$, and for any weight $\lambda$, denote

[TABLE]

• $\star$

We will associate to $\lambda$ a weight diagram $d_{\lambda}$, defined as a labeling of the integer line by symbols $\bullet$ (“black ball”) and $\circ$ (“empty”) such that such that $j$ has label $\bullet$ if $j\in\{\bar{\lambda_{i}}\,|\,i=1,2,\ldots\}$, and label $\circ$ otherwise.

• $\star$

We denote $\left|{\lambda}\right|:=-\sum_{i}{\lambda_{i}}$ and $\kappa(\lambda)=\sum_{i}(-1)^{\bar{\lambda}_{i}}$.

2.3.3. Representations of

$\mathfrak{p}(n)$

We denote by ${\mathcal{F}}_{n}$ the category of finite-dimensional representations of $\mathfrak{p}(n)$ whose restriction to $\mathfrak{p}(n)_{\bar{0}}\cong\mathfrak{gl}(n)$ integrates to an action of $GL(n)$.

By definition, the morphisms in ${\mathcal{F}}_{n}$ will be grading-preserving $\mathfrak{p}(n)$-morphisms, i.e., $\operatorname{Hom}_{{\mathcal{F}}_{n}}(X,Y)$ is a vector space and not a vector superspace. This is important in order to ensure that the category ${\mathcal{F}}_{n}$ be abelian.

The category ${\mathcal{F}}_{n}$ is not semisimple. In fact, this category is a highest-weight category, having simple, standard, costandard, and projective modules (these are also injective and tilting, per [BKN10]). Given a simple module $L_{n}(\lambda)$ in ${\mathcal{F}}_{n}$, we denote the corresponding standard, costandard, and projective modules by $\Delta_{n}(\lambda)$, $\nabla_{n}(\lambda)$, $P_{n}(\lambda)$ respectively.

2.3.4. Tensor Casimir and translation functors

Consider the following natural endomorphism ${\Omega}^{(n)}$ of the endofunctor $(-)\otimes V_{n}$ on $\mathcal{F}_{n}$.

Note that $\mathfrak{p}_{n}$ is the set of fixed points of the involutive automorphism $\sigma$ of $\mathfrak{gl}(V_{n})$. We consider the $\mathfrak{p}_{n}$-equivariant decomposition:

[TABLE]

where $\mathfrak{p}_{n}^{*}$ is the eigenspace of $\sigma$ with eigenvalue $-1$. Both $\mathfrak{p}_{n}$ and $\mathfrak{p}_{n}^{*}$ are maximal isotropic subspaces with respect to the invariant symmetric form on $\mathfrak{gl}(V_{n})$ and hence this form defines a non-degenerated pairing $\mathfrak{p}_{n}^{*}\otimes\mathfrak{p}_{n}\to{\mathbb{C}}$.

We begin by taking the orthogonal $\mathfrak{p}_{n}$-equivariant decomposition

[TABLE]

with respect to the form

[TABLE]

Definition 2.3.4** (Tensor Casimir).**

For any $M\in\mathcal{F}_{n}$, let ${\Omega}_{M}$ be the composition

[TABLE]

where $i_{*}:\mathfrak{p}(n)^{*}\to\mathfrak{gl}(V_{n})$ is the $\mathfrak{p}(n)$-equivariant embedding defined above.

Definition 2.3.5** (Translation functors).**

For $k\in{\mathbb{C}}$, we define a functor ${\Theta^{\prime}}^{(n)}_{k}:\mathcal{F}_{n}\to\mathcal{F}_{n}$ as the functor $\Theta^{(n)}=(-)\otimes V_{n}$ followed by the projection onto the generalized $k$-eigenspace for ${\Omega}^{(n)}$, i.e.

[TABLE]

and set ${\Theta}^{(n)}_{k}:=\Pi^{k}{\Theta^{\prime}}^{(n)}_{k}$ in case $k\in\mathbb{Z}$ (it was proved in [BDE*+*16] that $\forall k\notin{\mathbb{Z}},\;{\Theta}^{(n)}_{k}\cong 0$).

We use the following results from [BDE*+*16] throughout the paper:

Theorem 2.3.6** (See [BDE*+*16].).**

The relations on the translation functors $\Theta_{j}$, $j\in{\mathbb{Z}}$ induce a representation of the infinite Temperley-Lieb algebra $TL_{\infty}(q=i)$ on the Grothendieck ring on ${\mathcal{F}}_{n}$. Furthermore, for any $k\in{\mathbb{Z}}$, ${\Theta}^{(n)}_{k}\vdash{\Theta}^{(n)}_{k-1}$.

The functors $\Theta_{k}$ are exact, since $-\otimes V_{n}$ is an exact functor.

Theorem 2.3.7** (See [BDE*+*16].).**

Let $P$ be an indecomposable projective module in ${\mathcal{F}}_{n}$. Then for any $i$, $\Theta_{i}P$ is indecomposable projective or zero.

For more details on the structure of ${\mathcal{F}}_{n}$ we refer the reader to [BDE*+*16].

2.3.5. Blocks

There are $2(n+1)$ blocks in the category ${\mathcal{F}}_{n}$ has blocks. These blocks are in bijection with the set $\{-n,-n+2,\ldots,n-2,n\}\times\{+,-\}$.

We have a decomposition

[TABLE]

where the functor $\Pi$ (parity change) induces an equivalence $\left({\mathcal{F}}_{n}\right)_{k}^{+}\cong\left({\mathcal{F}}_{n}\right)_{k}^{-}$. The block $\left({\mathcal{F}}_{n}\right)_{k}$ contains all simple modules $L(\lambda)$ with $\kappa(\lambda)=k$.

Hence we may define up-to-pairity blocks

[TABLE]

By abuse of terminology, we will just call these “blocks” throughout the paper.

Theorem 2.3.8** (See [BDE*+*16].).**

Let $i\in\mathbb{Z}$, $k\in\{-n,-n+2,\ldots,n-2,n\}$. Then we have

[TABLE]

Finally, we introduce some notation:

Notation 2.3.9*.*

Let $I=(i_{1},i_{2},\ldots,i_{k})$ be a sequence with $i_{1},i_{2},\ldots,i_{k}\in{\mathbb{Z}}$.

1. (1)

We denote by

[TABLE]

the composition of the corresponding translation functors. 2. (2)

We set

[TABLE]

The following is an immediate corollary of Theorem 2.3.8:

Corollary 2.3.10**.**

For any $n\geq 1$, $l\in\{-n,-(n-2),\ldots,n-2,n\}$ and any integer sequence $I=(i_{1},i_{2},\ldots,i_{k})$ we have:

[TABLE]

2.4. The Duflo-Serganova functor

2.4.1. Definition and basic properties

Let $n\geq 3$, and let $x\in\mathfrak{p}(n)$ be an odd element such that $[x,x]=0$. We define the following correspondence of vector superspaces:

Definition 2.4.1** (See [DS05]).**

Let $M\in{\mathcal{F}}_{n}$. We define

[TABLE]

The vector superspace $\mathfrak{p}_{x}:=DS_{x}\mathfrak{p}(n)$ is naturally equipped with a Lie superalgebra structure. One can check by direct computations that $\mathfrak{p}_{x}$ is isomorphic to $\mathfrak{p}(n-s)$ where $s$ is the rank of $x$. The above correspondence defines an SM-functor $DS_{x}:{\mathcal{F}}_{n}\to{\mathcal{F}}_{n-s}$, called the Duflo-Serganova functor. Such functors were introduced in [DS05].

3. The Duflo-Serganova functor and the tensor Casimir

Let $n\geq 3$, and let $x\in\mathfrak{p}(n)_{\bar{1}}$ be such that $[x,x]=0$. Let $s:={\operatorname{rk}}(x)$.

Definition 2.4.1 then gives us a functor $DS_{x}:{\mathcal{F}}_{n}\to{\mathcal{F}}_{n-s}$.

Lemma 3.0.1**.**

We have: $DS_{x}({\Omega})={\Omega}^{(n-s)}DS_{n}$, where ${\Omega}^{(n)}$ is the tensor Casimir for $\mathfrak{p}(n)$, and $\Omega_{n-s}$ is the tensor Casimir for $\mathfrak{p}(n-s)$.

That is, for any $M\in\mathcal{F}_{n}$, $DS_{x}({\Omega}^{(n)}_{M})={\Omega}^{(n-s)}_{DS_{x}(M)}$ as endomorphisms of $V_{n-s}\otimes DS_{x}(M)$.

Proof.

This follows directly from the definition of the tensor Casimir (Definition 2.3.4), as well as the fact that $DS_{x}$ is a SM functor. ∎

Corollary 3.0.2**.**

The functor $DS_{x}$ commutes with translation functors, that is we have a natural isomorphism of functors

[TABLE]

for any $k\in{\mathbb{Z}}$.

Proof.

Recall that $DS_{x}$ is a SM functor and $DS_{x}(V_{n})\cong V_{n-s}$. Hence we have a natural isomorphism $\eta:DS_{x}\Theta\longrightarrow\Theta DS_{x}$, where $\Theta^{(n)}=(-)\otimes V_{n}$ is as in Definition 2.3.5.

Now, consider $DS_{x}\left(\Omega_{n}\right)$ (the tensor Casimir). By Lemma 3.0.1, the diagram below commutes:

[TABLE]

Hence $\eta$ induces an isomorphism $DS_{x}\Theta^{(n)}_{k}\cong\Theta^{(n-s)}_{k}DS_{x}$ for any $k\in{\mathbb{Z}}$, as required.

∎

4. Main result

Throughout this section, we will work with functors $DS_{x}:{\mathcal{F}}_{n}\to{\mathcal{F}}_{n-2}$ where $x\in\in\mathfrak{p}(n)_{-1}$ has rank $2$ and satisfies $[x,x]=0$. Here $\mathfrak{p}(n)_{-1}$ is as in Section 2.3.1.

Theorem 4.0.1**.**

We have ${\operatorname{sdim}}M=0$ if $M\in{\mathcal{F}}^{k}_{n}$, $k\neq 0,\pm 1$.

Proof.

Let $x\in\mathfrak{p}(n)_{-1}$ such that $[x,x]=0$, ${\operatorname{rk}}(x)=2$.

We prove the statement by induction on $n$.

The base case $n=1$ is tautological (in that case, there are no non-zero blocks except ${\mathcal{F}}^{1}_{\pm 1}$); in the base case $n=2$, it is enough to check for simple $M\in{\mathcal{F}}^{2}_{\pm 2}$. The blocks ${\mathcal{F}}^{2}_{\pm 2}$ are so-called typical blocks, hence all the simple objects in ${\mathcal{F}}^{2}_{\pm 2}$ are costandard ($L(\mu)=\nabla(\mu)$), and hence have dimension zero (cf. [BDE*+*16]).

Next, for the inductive step, let $n\geq 3$, and assume that our statement holds for $n-2$.

Let $M\in{\mathcal{F}}^{k}_{n}$, $k\notin\{0,1\}$. We use Proposition 4.0.2 below, which states that $DS_{x}$ preserves blocks, to show that $DS_{x}(M)$ lies in the corresponding block ${\mathcal{F}}_{n-2}^{k}$ (if $n=k$, then $DS_{x}(M)=0$).

The fact that the functor $DS_{x}:{\mathcal{F}}_{n}\to{\mathcal{F}}_{n-2}$ is SM, and hence preserves dimensions, allows us to use the inductive assumption to show that ${\operatorname{sdim}}DS_{x}(M)=0$, and hence ${\operatorname{sdim}}M=0$, as required.

∎

Proposition 4.0.2**.**

Let $x\in\mathfrak{p}(n)_{-1}$ such that $[x,x]=0$, ${\operatorname{rk}}(x)=2$. Let $M\in{\mathcal{F}}^{k}_{n}$. Then $DS_{x}(M)\in{\mathcal{F}}_{n-2}^{k}$ (if $k=\pm n$, then $DS_{x}(M)=0$).

Proof.

The proposition is proved in several steps.

1. (1)

We prove that $DS_{x}$ commutes with translation functors. This is done in Lemma 3.0.2. 2. (2)

We prove that it is possible to translate any simple module into any typical block; that is, for any simple $L\in{\mathcal{F}}_{n}$,

• •

There exists an integer sequence $I=(i_{1},i_{2},\ldots,i_{s})$ such that

[TABLE]

• •

There exists an integer sequence$I=(i_{1},i_{2},\ldots,i_{s})$ such that

[TABLE]

This is proved in Lemma 4.0.3. 3. (3)

We prove the statement in the case $k=\pm n$. In that case, we need to show that $DS_{x}(M)=0$ for any $M\in{\mathcal{F}}^{\pm n}_{n}$. Recall from [BDE*+*16, Remark 9.1.3] that any simple module in such a block is also costandard (that is, a thin Kac module in the terminology of [BDE*+*16]). Hence it is a free $U(\mathfrak{p}(n)_{-1})$-module. Any $\mathfrak{p}(n)$-module in ${\mathcal{F}}^{\pm n}_{n}$ has a finite filtration with simple subquotients, hence is a free $U(\mathfrak{p}(n)_{-1})$-module. This implies that $DS_{x}(M)=0$ for any $M\in{\mathcal{F}}^{\pm n}_{n}$. 4. (4)

Consider a simple module $L\in{\mathcal{F}}^{k}_{n}$, and a simple subquotient $L^{\prime}$ of $DS_{x}(L)$. Let $l$ be such that $L^{\prime}\in{\mathcal{F}}_{n-2}^{l}$.

We will show that $k=l$.

Assume $l<k$. Recall that $k\equiv l\,(\mathrm{mod}\,2)$, as explained in Section 2.3.5.

By (2), there exists an integer sequence $I:=(i_{1},i_{2},\ldots,i_{s})$ such that the translation functor

[TABLE]

on ${\mathcal{F}}_{n-2}$ satisfies:

[TABLE]

Furthermore, by Corollary 2.3.10, we have: $t(I)=(n-2-l)/2$.

Let

[TABLE]

be the corresponding translation functor on ${\mathcal{F}}_{n}$.

By (1), we have an isomorphism

[TABLE]

By our construction, this object has a non-zero direct summand in the typical block ${\mathcal{F}}_{n-2}^{n-2}$.

Let us show that $\Theta^{(n)}_{I}L\in{\mathcal{F}}_{n}^{n}$. Indeed, we may apply Corollary 2.3.8 to get:

[TABLE]

We already computed that $t(I)=(n-2-l)/2$ hence ${\mathcal{F}}_{n}^{k+2t(I)}={\mathcal{F}}_{n}^{n+(k-l)-2}$. Recall that $k>l$ and they have the same parity, so $n+(k-l)-2\geq n$. Now, if $n+(k-l)-2>n$, then ${\mathcal{F}}_{n}^{n+(k-l)-2}=0$, so we can just say that $\Theta^{(n)}_{I}L\in{\mathcal{F}}_{n}^{n}$.

This implies that $DS_{x}\Theta^{(n)}_{I}L=0$ (by (3)), and hence cannot have a non-zero direct summand in the typical block ${\mathcal{F}}_{n-2}^{n-2}$. Thus we obtained a contradiction.

A similar proof shows that we cannot have $l>k$: in that case, we to translate to typical block ${\mathcal{F}}_{n-2}^{-(n-2)}$.

∎

Lemma 4.0.3**.**

For any simple module $L\in{\mathcal{F}}_{n}$,

1. (1)

There exists a composition of translation functors $\Theta_{I}$ where $I=(i_{1},i_{2},\ldots,i_{s})$ is an integer sequence, such that $\Theta^{(n)}_{I}L\neq 0$ sits in the typical block ${\mathcal{F}}_{n}^{n}$. 2. (2)

There exists a composition of translation functors $\Theta_{I}$ where $I=(i_{1},i_{2},\ldots,i_{s})$ is an integer sequence, such that $\Theta^{(n)}_{I}L\neq 0$ sits in the typical block ${\mathcal{F}}_{n}^{-n}$.

Proof.

We use the results of [BDE*+*16] on the action of translation functors. In particular, we use the description action of translation functors on projective modules given in [BDE*+*16, Section 7.2] as well as the adjuction $\Theta^{(n)}_{i}\vdash\Theta^{(n)}_{i-1}$ for any $i\in{\mathbb{Z}}$.

We first prove (1).

Let $\lambda$ be the highest weight of $L$ (hence $L=L_{n}(\lambda)$), let $P=P_{n}(\lambda)$ be the projective cover of $L$.

Fix a typical weight $\mu$ of $\mathfrak{p}(n)$ with $\mu_{i}<\lambda_{i}$ and $\mu_{i}\in 2{\mathbb{Z}}$ for all $i$. Such a weight clearly exists: take for example $\mu_{n}\in{\mathbb{Z}}$ such that $\mu_{n}-n\in 2{\mathbb{Z}}$ and $\mu_{n}<\lambda_{1}$. Set $\mu_{k}:=\mu_{n}-(n-k)$ for any $k=1,2,\ldots,n-1$. Then $\mu_{k}<\mu_{n}<\lambda_{1}<\lambda_{k}$ for any $k=1,\ldots,n$, and

[TABLE]

which implies $\mu_{k}+k\in 2{\mathbb{Z}}$. Hence $\mu$ is a typical weight, and $P^{\prime}:=P_{n}(\mu)$ sits in ${\mathcal{F}}_{n}^{n}$.

By [BDE*+*16, Section 7.2], we have an integer sequence $J=(j_{1},j_{2},\ldots,j_{s})$ such that $\Theta^{(n)}_{J}P^{\prime}=P$.

Set $I=(j_{s}-1,j_{s-1}-1,\ldots,j_{2}-1,j_{1}-1)$. Then $\Theta^{(n)}_{J}\vdash\Theta^{(n)}_{I}$, and we have:

[TABLE]

Therefore $\Theta^{(n)}_{I}L$ is a non-trivial quotient of $P^{\prime}$. This proves (1).

Similarly, we prove (2). Fix a typical weight $\mu$ of $\mathfrak{p}(n)$ with $\mu_{i}<\lambda_{i}$ and $\mu_{i}\in 2{\mathbb{Z}}+1$ for all $i$. Again, such a weight can be constructed very explicitly. Then $P^{\prime}:=P_{n}(\mu)$ sits in ${\mathcal{F}}_{n}^{-n}$, and we can apply exact the same arguments as before.

∎

5. Dual modules and blocks

Proposition 5.0.1**.**

Let $M\in{\mathcal{F}}_{n}^{k}$. Then $M^{*}$ also lies in block ${\mathcal{F}}_{n}^{k}$.

Proof.

First of all, notice that it is enough to prove the statement for a simple module $M=L(\lambda)\in{\mathcal{F}}_{n}^{k}$.

Consider the costandard module $\nabla(\lambda)$ having $L(\lambda)$ as its socle. This module is indecomposable, so $\nabla(\lambda)\in{\mathcal{F}}_{n}^{k}$. Consider the dual module $\nabla(\lambda)^{*}$. This is also an indecomposable costandard module, with cosocle $L(\lambda)^{*}$, so it is enough to check that $\nabla(\lambda)^{*}\in{\mathcal{F}}_{n}^{k}$ as well. Now, by [BDE*+*16, Lemma 3.6.1], $\nabla(\lambda)^{*}\cong\nabla(\mu)$, where $\mu+\rho=-w_{0}({\lambda}+\rho)$, where $w_{0}$ is the longest element in the Weyl group. That is, $d_{\mu}$ is obtained from $d_{{\lambda}}$ by reflecting the diagram with respect to zero.

Hence $\kappa({\lambda})=\kappa(\mu)$, and so $L(\mu)\in{\mathcal{F}}_{n}^{k}$. ∎

Proposition 5.0.2**.**

There exists a natural isomorphism

[TABLE]

Proof.

Consider the functor $-\otimes V_{n}:{\mathcal{F}}_{n}\to{\mathcal{F}}_{n}$, $M\mapsto M\otimes V_{n}$. We have natural isomorphisms

[TABLE]

where $\eta:V_{n}^{*}\to V_{n}\otimes\Pi{\mathbb{C}}$ is the isomorphism defined by the odd bilinear form on $V_{n}$.

Consider the tensor Casimir $\Omega_{M}:M\otimes V_{n}\to M\otimes V_{n}$ as in Definition 2.3.4. Choosing dual bases $\{X_{i}\}_{i},\{X^{i}\}_{i}$ in $\mathfrak{p}(n)$ and $\mathfrak{p}(n)^{*}\subset\mathfrak{gl}(n|n)$, we can write $\Omega_{M}=\sum_{i}X_{i}\rvert_{M}\otimes X^{i}\rvert_{V_{n}}$.

Denote by

[TABLE]

the dual map. Then for any homogeneous $u\in V_{n}^{*},f\in M^{*}$, we have:

[TABLE]

We now construct the commutative diagram

[TABLE]

and we compute the lower two horizontal arrows.

We begin with the horizontal arrow $\phi:M^{*}\otimes V_{n}^{*}\to M^{*}\otimes V_{n}^{*}$.

By definition, $\phi=\sigma_{M^{*},V_{n}^{*}}^{-1}\circ\Omega_{M}^{*}\circ\sigma_{M^{*},V_{n}^{*}}$.

For any homogeneous $f\in M^{*},u\in V_{n}^{*}$, applying the map $\Omega_{M}^{*}\circ\sigma_{M^{*},V_{n}^{*}}$ to the element $f\otimes u$ we get:

[TABLE]

Hence

[TABLE]

Next, we compute the horizontal arrow $\phi^{\prime}:M^{*}\otimes V_{n}\otimes\Pi{\mathbb{C}}\to M^{*}\otimes V_{n}\otimes\Pi{\mathbb{C}}$.

The elements $X^{i}\in\mathfrak{gl}(n|n)$ satisfy the following property (cf. [BDE*+*16, Proof of Proposition 4.4.1]):

[TABLE]

Given homogeneous $f\in M^{*},v\in V_{n}\otimes\Pi{\mathbb{C}}$, we have

[TABLE]

Hence $\phi^{\prime}=-\Pi\Omega_{M^{*}}$. Thus the natural isomorphism $\sigma_{M*,V_{n}^{*}}\circ(\operatorname{Id}\otimes\eta^{-1})$ establishes a natural isomorphism between the eigenspace of $\Omega_{M}^{*}$ corresponding to eigenvalue $k$ and the eigenspace (shifted by $\Pi$) of $\Omega_{M^{*}}=-\Pi\phi^{\prime}$ corresponding to eigenvalue $(-k)$. This implies the statement of the proposition. ∎

Example 5.0.3**.**

Let $n\geq 2$ and set $M=V_{n}$. Then $V_{n}^{*}\cong\Pi V_{n}$ and $\Theta_{-1}V_{n}\cong\wedge^{2}V_{n}$ and hence

[TABLE]