Schur Superalgebras Research Articles

Rational Schur Superalgebras

We develop and study the generalization of rational Schur algebras to the super setting. Similar to the classical case, this provides a new method for studying rational supermodules of the general linear supergroup [Formula: see text]. Furthermore, we establish a Schur-Weyl duality result for rational Schur superalgebras and conclude that under certain conditions these objects will be semisimple.

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.

Superized Troesch complexes and cohomology for strict polynomial superfunctors

We adapt a construction due to Troesch to the category of strict polynomial superfunctors in order to construct complexes of injective objects whose cohomology is isomorphic to Frobenius twists of the (super)symmetric power functors . We apply these complexes to construct injective resolutions of the even and odd Frobenius twist functors, to investigate the structure of the Yoneda algebra of the Frobenius twist functor, and to compute other extension groups between strict polynomial superfunctors. By an equivalence of categories, this also provides cohomology calculations in the category of left modules over Schur superalgebras.

Levi-type Schur–Sergeev duality for general linear super groups

In this paper, we investigate a kind of double centralizer property for general linear supergroups. For the super space [Formula: see text] over an algebraically closed field [Formula: see text] whose characteristic is not equal to [Formula: see text], we consider its [Formula: see text]-homogeneous one-dimensional extension [Formula: see text], and the natural action of the supergroup [Formula: see text] on [Formula: see text]. Then we have the tensor product supermodule ([Formula: see text], [Formula: see text]) of [Formula: see text]. We present a kind of generalized Schur–Sergeev duality which is said that the Schur superalgebras [Formula: see text] of [Formula: see text] and the so-called weak degenerate double Hecke algebra [Formula: see text] are double centralizers. The weak degenerate double Hecke algebra is an infinite-dimensional algebra, which has a natural representation on the tensor product space. This notion comes from [B. Shu, Y. Xue and Y. Yao, On enhanced reductive groups (I): Parabolic Schur algebras and the dualities related to degenerate double Hecke algebras, preprint (2013), arXiv:2005.13152 [Math. RT]], with a little modification.

A REALIZATION OF THE ENVELOPING SUPERALGEBRA

AbstractIn [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum $\mathfrak {gl}_n$ via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ of the loop algebra $\widehat {\mathfrak {gl}}_{m|n}$ of ${\mathfrak {gl}}_{m|n}$ with those of affine symmetric groups ${\widehat {{\mathfrak S}}_{r}}$ . Then, we give a BLM type realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ via affine Schur superalgebras.The first application of the realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ is to determine the action of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ on tensor spaces of the natural representation of $\widehat {\mathfrak {gl}}_{m|n}$ . These results in epimorphisms from $\;{\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ to affine Schur superalgebras so that the bridging relation between representations of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ and ${\widehat {{\mathfrak S}}_{r}}$ is established. As a second application, we construct a Kostant type $\mathbb Z$ -form for ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur–Weyl duality in arbitrary characteristics.

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.

MULTIPLICATION FORMULAS AND SEMISIMPLICITY FOR -SCHUR SUPERALGEBRAS

We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for the $q$-Schur superalgebras. This gives a combinatorialization of the relative norm approach developed in Du and Gu (A realization of the quantum supergroup$\mathbf{U}(\mathfrak{g}\mathfrak{l}_{m|n})$, J. Algebra 404 (2014), 60–99). We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for $q$-Schur superalgebras. We also construct infinitesimal and little $q$-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.

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.

The BLM realization for the integral form of Uv(glm|n)

In this paper, we embed the integral form of the quantum supergroup ${\boldsymbol~U}_\bv(\mathfrak{gl}_{m\,|\,n})$ to the product of a family of integral quantum Schur superalgebras. We show that the image of the embedding is a free $\mathbb{Z}[{\boldsymbol~v},{\boldsymbol~v}^{-1}]$-module by finding the basis explicitly and calculating the fundamental multiplication formulas of these bases. Unlike the non-super case, the fundamental multiplication formula, which is the key step, is more complicated since we have to deal with the case of multiplying the odd root vectors. As a consequence, via the base change, we realize the quantum supergroup at roots of unity as a subalgebra of the product of quantum Schur superalgebras. Thus, we find a new basis of quantum supergroups at odd roots of unity which comes from quantum Schur superalgebras.

Derived Representation Type of Schur Superalgebras

In this note we extend the results of Bekkert and Futorny in [2] and Hemmer, Kujawa and Nakano in [10] and determine the derived representation type of Schur superalgebras.

A realisation of the quantum supergroup [formula omitted

We reconstruct the quantum enveloping superalgebra U(glm|n) over Q(υ) via (finite dimensional) quantum Schur superalgebras. In particular, we obtain a new basis containing the standard generators for U(glm|n) and explicit multiplication formulas of an arbitrary basis element by a generator.

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.

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.

Some properties of general linear supergroups and of Schur superalgebras

It is shown that the category of rational supermodules over a general linear supergroup is a highest weight category. More exactly, we construct superanalogs of the theory of modules with good filtration and of the dual theory of modules with Weyl’s. Using these, we show that indecomposable injective supermodules have good filtration of a certain kind.

Exactness of Complexes of Modules over Schur Superalgebras

In this paper, we study tensor products of injective modules and exactness of certain naturally appearing complexes of projective and injective modules over Schur superalgebras over an algebraically closed field K of characteristic p > 2. In particular, for the complex [Formula: see text] of projective modules P(i) = Sξ 1i(m+1)r-i given by multiplication by ui = ξ 1i(m+1)r-i, 1i+1 (m+1)r-i-1, we determine all of the terms where it is exact.

Schur Superalgebras in Characteristic p

The structure of a Schur superalgebra S = S(1 ∣ 1, r) in odd characteristic p is completely determined. The algebra S is semisimple if and only if p does not divide r. If p divides r, then simple S-modules are one-dimensional and the quiver and relations of S can be immediately seen from its regular representation computed in this paper. Surprisingly, if p divides r, then S is neither quasi-hereditary nor cellular nor stratified, as one would expect by analogy with classical Schur algebras or Schur superalgebras in characteristc zero.

Borel Subalgebras of Schur Superalgebras

It is proved that any Schur superalgebra is representable as a product of two Borel subalgebras of that superalgebra, which are symmetric w.r.t. its natural anti-isomorphism (Bruhat-Tits decomposition). This readily implies that any simple module is uniquely defined by its highest weight, and all other weights are strictly less than is the highest under the dominant ordering. It is stated that the fundamental theorem of Kempf, which is valid for all classical Schur algebras, might be true for superalgebras only if they are semisimple. Nevertheless, a weaker theorem of Grothendieck holds true for superalgebras since Borel subalgebras are quasihereditary. Also we formulate an analog of the Donkin-Mathieu theorem for Schur superalgebras, and show that it is valid in the elementary non-classical case, that is, for the algebras S(1|1, r).