[Paper Review] The queer q-Schur superalgebra

Jie Du, Jinkui Wan|arXiv (Cornell University)|Nov 17, 2015

Algebraic structures and combinatorial models20 references3 citations

TL;DR

This paper introduces the queer $q$-Schur superalgebra as the endomorphism algebra of induced $q$-permutation supermodules over the Hecke–Clifford superalgebra, constructs an integral basis using matrices and circled tableaux, establishes the base change property, and identifies the superalgebra with the quantum queer Schur superalgebra. It provides a complete classification of irreducible polynomial superrepresentations via a Drinfeld–Jimbo type presentation.

ABSTRACT

As a natural generalization quantum Schur algebras associated with the Hecke algebra of the symmetric group, we introduce the quantum Schur superalgebra of type Q associated with the Hecke-Clifford superalgebra, which, by definition, is the endomorphism algebra of the induced module over the Hecke-Clifford superalgebra from certain permutation modules over the Hecke algebra of the symmetric group. 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 quantum Schur superalgebra of type Q with the quantum queer Schur superalgebras investigated in the context of quantum queer supergroups and then provide a classification of their irreducible representations over a certain extension of the field of complex rational functions.

Motivation & Objective

• To generalize Green's standard treatment of $q$-Schur algebras to a new class of Schur superalgebras, specifically the queer $q$-Schur superalgebra.
• To define the queer $q$-Schur superalgebra as the endomorphism algebra of induced $q$-permutation supermodules over the Hecke–Clifford superalgebra.
• To construct an integral basis for the queer $q$-Schur superalgebra using matrix and tableau combinatorics.
• To establish the base change property for the queer $q$-Schur superalgebra over arbitrary commutative rings.
• To identify the queer $q$-Schur superalgebra with the quantum queer Schur superalgebra and classify its irreducible polynomial superrepresentations in the generic case.

Proposed method

• Define the queer $q$-Schur superalgebra $\mathcal{Q}_q(n,r;R)$ as the endomorphism algebra of the induced $\mathcal{H}^\text{\sf c}_{r,R}$-module from $q$-permutation supermodules over $\mathcal{H}_{r,R}$.
• Construct a standard basis using double coset analysis in the symmetric group, with basis elements labeled by matrices $A|B$ in $M(\mathbb{N}|\mathbb{Z}_2)_{\mu,\lambda}$.
• Use circled row-semistandard tableaux and formal linear combinations with $q$-powers to define elements $m_{\mathsf{S}}$ in the endomorphism algebra.
• Establish the base change property by showing compatibility with base extension from $\mathbb{Z}[\boldsymbol{q}]$ to arbitrary commutative rings $R$.
• Identify the queer $q$-Schur superalgebra with the quantum queer Schur superalgebra via a Drinfeld–Jimbo type presentation.
• Apply results from Jones and Nazarov to construct a complete set of irreducible polynomial superrepresentations over an extension of $\mathbb{C}(\boldsymbol{v})$.

Experimental results

Research questions

• RQ1 How can the $q$-Schur algebra construction be generalized to the superalgebra setting, particularly for the queer $q$-Schur superalgebra?
• RQ2 What is the structure of the endomorphism algebra of induced $q$-permutation supermodules over the Hecke–Clifford superalgebra?
• RQ3 Can an integral basis be constructed for the queer $q$-Schur superalgebra using matrix and tableau combinatorics?
• RQ4 Does the queer $q$-Schur superalgebra satisfy the base change property over arbitrary commutative rings?
• RQ5 How can the irreducible polynomial superrepresentations of the quantum queer supergroup be classified?

Key findings

• The queer $q$-Schur superalgebra $\mathcal{Q}_q(n,r;R)$ is isomorphic to the quantum queer Schur superalgebra defined via a Drinfeld–Jimbo presentation.
• An integral basis for $\mathcal{Q}_q(n,r;R)$ is constructed using matrices $A|B$ with entries in $\mathbb{N}$ and $\mathbb{Z}_2$, corresponding to uncircled and circled entries in tableaux.
• The basis elements $m_{\mathsf{S}}$ are defined via formal linear combinations over $q$-powers and tableaux with fixed circle positions, ensuring compatibility with the algebra structure.
• The base change property holds: the superalgebra over $R$ is obtained by base change from the universal $\mathbb{Z}[\boldsymbol{q}]$-algebra.
• The irreducible polynomial superrepresentations of the quantum queer supergroup are completely classified using the identification with the queer $q$-Schur superalgebra and results from Jones and Nazarov.
• The identification removes the need for the assumption that $1+q$ is not a zero-divisor in $R$, as shown in Remark A.3.