Schur Superalgebra Research Articles

Multiplication formulas and isomorphism theorem of ıSchur superalgebras

We introduce the notion of ı Schur superalgebra, which can be regarded as a type B/C counterpart of the q -Schur superalgebra (of type A) formulated as centralizer algebras of certain signed q -permutation modules over Hecke algebras. Some multiplication formulas for ı Schur superalgebra are obtained to construct their canonical bases. Furthermore, we established an isomorphism theorem between the ı Scuhr superalgebras and the q -Schur superalgebras of type A, which helps us derive semisimplicity criteria of the ı Schur superalgebras.

Symmetrizers for Schur superalgebras

For the Schur superalgebra S = S ( m | n , r ) over a ground field K of characteristic zero, we define symmetrizers T λ [ i : j ] of the ordered pairs of tableaux ( T i , T j ) of the shape λ . We show that the K -span A λ , K of all symmetrizers T λ [ i : j ] has a basis consisting of T λ [ i : j ] for T i and T j semistandard. In particular, A λ , K ≠ 0 if and only if λ is an ( m | n ) -hook partition. In this case, the S -superbimodule A λ , K is identified as D λ ⊗ K D λ o , where D λ and D λ o are left and right irreducible S -supermodules of the highest weight λ . We define modified symmetrizers T λ { i : j } and show that their Z -span forms a Z -form A λ , Z of A λ , Q . We show that every modified symmetrizer T λ { i : j } is a Z -linear combination of symmetrizers T λ { i : j } for T i , T j semistandard. Using modular reduction to a field K of characteristic p > 2 , we obtain that A λ , K has a basis consisting of modified symmetrizers T λ { i : j } for T i and T j semistandard.

The integral Schur-Weyl-Sergeev duality

Degenerating the quantum queer Schur superalgebra ${\mathcal{Q}_q(n,r; R)}$ to the case $q=1$, the queer Schur superalgebra ${\mathcal{Q}(n,r)}$ is obtained. In this article, we reconstruct the universal enveloping algebra ${U({\mathfrak{q}_n})}$ of the queer Lie superalgebra ${\mathfrak{q}_n}$ via ${\mathcal{Q}(n,r)}$, and achieve another explanation of the Schur-Weyl-Sergeev duality. Finally, we depict the Schur-Weyl-Sergeev duality over $\mathbb{Z}$.

Cyclotomic q-Schur superalgebras

The paper aims to introduce the cyclotomic [Formula: see text]-Schur superalgebra via the permutation supermodules of the cyclotomic Hecke algebra and investigate its structure. In particular, we show that the cyclotomic [Formula: see text]-Schur superalgebra is a cellular superalgebra and establish the double centralizer property between the cyclotomic Hecke algebra and the cyclotomic [Formula: see text]-Schur superalgebra.

DONKIN–KOPPINEN FILTRATION FOR GL(m|n) AND GENERALIZED SCHUR SUPERALGEBRAS

The paper contains results that characterize the Donkin–Koppinen filtration of the coordinate superalgebra K[G] of the general linear supergroup G = GL(m|n) by its subsupermodules CΓ = OΓ(K[G]). Here, the supermodule CΓ is the largest subsupermodule of K[G] whose composition factors are irreducible supermodules of highest weight ⋋, where ⋋ belongs to a finitely-generated ideal Γ of the poset X(T)+ of dominant weights of G. A decomposition of G as a product of subsuperschemes U–×Gev×U+ induces a superalgebra isomorphism ϕ* K[U–]⊗K[Gev]⊗K[U+]≃K[G]. We show that CΓ=ϕ*(K[U–]⊗MΓK[U+]), where MΓ=OΓ(K[Gev]). Using the basis of the module MΓ, given by generalized bideterminants, we describe a basis of CΓ. Since each CΓ is a subsupercoalgebra of K[G], its dual $$ {C}_{\Gamma}^{\ast }={S}_{\Gamma} $$ is a (pseudocompact) superalgebra called the generalized Schur superalgebra. There is a natural superalgebra morphism πΓ : Dist(G) → SΓ such that the image of the distribution algebra Dist(G) is dense in SΓ. For the ideal $$ X{(T)}_l^{+}, $$ of all weights of fixed length l, the generators of the kernel of $$ {\uppi}_{X{(T)}_l^{+}} $$ are described.

Quantized enveloping superalgebra of type P

We introduce a new quantized enveloping superalgebra $\mathfrak{U}_q{\mathfrak{p}}_n$ attached to the Lie superalgebra ${\mathfrak{p}}_n$ of type $P$. The superalgebra $\mathfrak{U}_q{\mathfrak{p}}_n$ is a quantization of a Lie bisuperalgebra structure on ${\mathfrak{p}}_n$ and we study some of its basic properties. We also introduce the periplectic $q$-Brauer algebra and prove that it is the centralizer of the $\mathfrak{U}_q {\mathfrak{p}}_n$-module structure on ${\mathbb C}(n|n)^{\otimes l}$. We end by proposing a definition for a new periplectic $q$-Schur superalgebra.

Even-primitive vectors in induced supermodules for general linear supergroups and in costandard supermodules for Schur superalgebras

Let $$G=GL(m|n)$$ be the general linear supergroup over an algebraically closed field K of characteristic zero, and let $$G_{ev}=GL(m)\times GL(n)$$ be its even subsupergroup. The induced supermodule $$H^0_G(\lambda )$$, corresponding to a dominant weight $$\lambda $$ of G, can be represented as $$H^0_{G_{ev}}(\lambda )\otimes \Lambda (Y)$$, where $$Y=V_m^*\otimes V_n$$ is a tensor product of the dual of the natural GL(m)-module $$V_m$$ and the natural GL(n)-module $$V_n$$, and $$\Lambda (Y)$$ is the exterior algebra of Y. For a dominant weight $$\lambda $$ of G, we construct explicit $$G_{ev}$$-primitive vectors in $$H^0_G(\lambda )$$. Related to this, we give explicit formulas for $$G_{ev}$$-primitive vectors of the supermodules $$H^0_{G_{ev}}(\lambda )\otimes \otimes ^k Y$$. Finally, we describe a basis of $$G_{ev}$$-primitive vectors in the largest polynomial subsupermodule $$\nabla (\lambda )$$ of $$H^0_G(\lambda )$$ (and therefore in the costandard supermodule of the corresponding Schur superalgebra S(m|n)). This yields a description of a basis of $$G_{ev}$$-primitive vectors in arbitrary induced supermodule $$H^0_G(\lambda )$$.

${\boldsymbol~q}$-Schur superalgebras

This paper provides a survey on the latest developments of $q$-Schur superalgebras, quantum linear supergroups and their canonical bases and irreducible (polynomial) representations. We first introduce the definitions of $q$-Schur superalgebras in three different contexts and their associated bases,and describe the relationships between the three bases. We then move on to display certain multiplication formulas in $q$-Schur superalgebras and discuss their applications to the new realizationof quantum linear supergroups and to the regular representation of a $q$-Schur superalgebra. We provide another application for the construction of canonical bases for the positive part of thequantum linear supergroup. As a by-product, a semi-simplicity criterion is given for $q$-Schur superalgebras. Moreover, by generalising the idea of Alperins weight conjecture and Scottspermutation representation theory, we provide a classification of irreducible modules for a $q$-Schur superalgebra. We also mention a new approach to introduce infinitesimal and little $q$-Schur superalgebras without using quantum coordinate superalgebras.

THE QUEER -SCHUR SUPERALGEBRA

As a natural generalisation of $q$-Schur algebras associated with the Hecke algebra ${\mathcal{H}}_{r,R}$ (of the symmetric group), we introduce the queer $q$-Schur superalgebra associated with the Hecke–Clifford superalgebra ${\mathcal{H}}_{r,R}^{\mathsf{c}}$, which, by definition, is the endomorphism algebra of the induced ${\mathcal{H}}_{r,R}^{\mathsf{c}}$-module from certain $q$-permutation modules over ${\mathcal{H}}_{r,R}$. We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base-change property for them. We will also identify the queer $q$-Schur superalgebras with the quantum queer Schur superalgebras investigated in the context of quantum queer supergroups and provide a constructible classification of their simple polynomial representations over a certain extension of the field $\mathbb{C}(\mathbf{v})$ of complex rational functions.

Description of costandard modules for Schur superalgebra S(2|2) in positive characteristic

In this paper, we shall describe costandard modules [Formula: see text] of restricted highest weight [Formula: see text] for Schur superalgebra [Formula: see text] over an algebraically closed field [Formula: see text] of positive characteristic [Formula: see text]. Additionally, for a restricted highest weight [Formula: see text], we determine all composition factors of the costandard module [Formula: see text]; in particular we compute the decomposition numbers in the process of the modular reduction of a simple module with a restricted highest weight.

On Schur Superfunctors

We introduce super-analogues of the Schur functors defined by Akin, Buchsbaum and Weyman. These Schur superfunctors may be viewed as characteristic-free analogues of finite dimensional irreducible polynomial representations of the Lie superalgebra 𝔤𝔩(m|n) studied by Berele and Regev. Our construction realizes Schur superfunctors as objects of a certain category of strict polynomial superfunctors. We show that Schur superfunctors are indecomposable objects of this category. In characteristic zero, these correspond to the set of all simple supermodules for the Schur superalgebra, S(m|n, d), for any m, n, d ⩾ 0. We also provide decompositions of Schur bisuperfunctors in terms of tensor products of skew Schur superfunctors.

Presenting Queer Schur Superalgebras

Associated to the two types of finite dimensional simple superalgebras, there are the general linear Lie superalgebra and the queer Lie superalgebra. The universal enveloping algebras of these Lie superalgebras act on the tensor spaces of the natural representations and, thus, define certain finite dimensional quotients, the Schur superalgebras and the queer Schur superalgebra. In this paper, we introduce the quantum analogue of the queer Schur superalgebra and investigate the presentation problem for both the queer Schur superalgebra and its quantum analogue.

Spin polynomial functors and representations of Schur superalgebras

We introduce categories of homogeneous strict polynomial functors, P o l d , k I \mathsf {Pol}^{\mathrm {I}}_{d,\Bbbk } and P o l d , k I I \mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk } , defined on vector superspaces over a field k \Bbbk of characteristic not equal 2. These categories are related to polynomial representations of the supergroups G L ( m | n ) GL(m|n) and Q ( n ) Q(n) . In particular, we prove an equivalence between P o l d , k I \mathsf {Pol}^{\mathrm {I}}_{d,\Bbbk } , P o l d , k I I \mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk } and the category of finite dimensional supermodules over the Schur superalgebra S ( m | n , d ) \mathcal {S}(m|n,d) , Q ( n , d ) \mathcal {Q}(n,d) respectively provided m , n ≥ d m,n \ge d . We also discuss some aspects of Sergeev duality from the viewpoint of the category P o l d , k I I \mathsf {Pol}^{\mathrm {II}}_{d,\Bbbk } .

Description of Costandard Modules of Schur Superalgebra S(3 | 1)

Let S(m | n) be a Schur superalgebra over a base field K of any characteristic different from 2. The characters of simple S(m | n)-modules and composition factors of costandard modules ∇(λ) for restricted weights λ were determined for the Schur superalgebra S(2 | 1) in [6] and for Schur superalgebra S(2 | 2) in [4] and [5]. The goal of this article is to obtain corresponding results for Schur superalgebra S(3 | 1). Additionally, we determine the character of ∇(λ) for all Schur superalgebras S(m | 1), where m ≥ 1.

Presenting Schur superalgebras

We provide a presentation of the Schur superalgebra and its quantum analogue which generalizes the work of Doty and Giaquinto for Schur algebras. Our results include a basis for these algebras and a presentation using weight idempotents in the spirit of Lusztig's modified quantum groups.

DESCRIPTION OF SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(2|2)

AbstractThe goal of this paper is to describe explicitly simple modules for Schur superalgebra S(2|2) over an algebraically closed field K of characteristic zero or positive characteristic p>2.

A note on bideterminants for Schur superalgebras

Let S ( m | n , r ) Z be a Z -form of a Schur superalgebra S ( m | n , r ) generated by elements ξ i , j . We solve a problem of Muir and describe a Z -form of a simple S ( m | n , r ) -module D λ , Q over the field Q of rational numbers, under the action of S ( m | n , r ) Z . This Z -form is the Z -span of modified bideterminants [ T ℓ : T i ] defined in this work. We also prove that each [ T ℓ : T i ] is a Z -linear combination of modified bideterminants corresponding to ( m | n ) -semistandard tableaux T i .

Description of costandard modules for Schur superalgebra S(2|1) in positive characteristic

We describe the characters of simple modules and composition factors of costandard modules for S(2|1) in positive characteristics and verify a conjecture of La Scala–Zubkov regarding polynomial superinvariants for GL(2|1).

COSTANDARD MODULES OVER SCHUR SUPERALGEBRAS IN CHARACTERISTIC p

In this paper we consider the problem of describing the costandard modules ∇(λ) of a Schur superalgebra S(m|n,r) over a base field K of arbitrary characteristic. Precisely, if G = GL(m|n) is a general linear supergroup and Dist (G) its distribution superalgebra we compute the images of the Kostant ℤ-form under the epimorphism Dist (G) → S(m|n,r). Then, we describe ∇(λ) as the null-space of some set of superderivations and we obtain an isomorphism ∇(λ) ≈ ∇(λ+|0) ⊗ ∇(0|λ-) assuming that λ = (λ+|λ-) and λm = 0. If char (K) = p we give a Frobenius isomorphism ∇(0|pμ) ≈ ∇(μ)p where ∇(μ) is a costandard module of the ordinary Schur algebra S(n,r). Finally we provide a characteristic free linear basis for ∇(λ|0) which is parametrized by a set of superstandard tableaux.

Representation type of Schur superalgebras

Abstract Let S(m|n, d) be the Schur superalgebra whose supermodules correspond to the polynomial representations of the supergroup GL(m|n) of degree d. In this paper we determine the representation type of these algebras (that is, we classify the ones which are semisimple, have finite, tame and wild representation type). Moreover, we prove that these algebras are in general not quasi-hereditary and have infinite global dimension.