Irreducible representations of q-Schur superalgebras at a root of unity

Abstract

Abstract. Under the assumption that the quantum parameter q is an lth primitive root of unity with l odd in a field F of characteristic 0 and m + n ≥ r, we obtained a complete classification of irreducible modules of the q-Schur superalgebra S F (m|n, r) introduced in [11].

Polynomial super representations of $$U_{q}^{\mathrm{res}}(\mathfrak {gl}_{m|n})$$ at roots of unity

“…(1) When m + n ≥ r, a classification is given in [10,11] without using representations of the quantum supergroup. See also a comparison of the index sets in [10,Theorem B.3] in this case. Note that the theorem above has also generalised the classification loc.…”

“…(3) In [10,11], a classification is done by using the defect groups of primitive idempotents. By (2), we see that the non-equivalent primitive idempotents e 1 , e 2 , · · · , e N can be selected to satisfy the condition e i 1 λ (i) = e i for every i.…”

The Queer -Schur Superalgebra

“…(2) We remark that, for the q-Schur superalgebras S v (m|n, r) F of type M with m + n ≥ r, their irreducible representations at (odd) roots of unity have been classified in [11], while a non-constructible classification of irreducible Q v (n, r) F -supermodules is obtained in [25,Theorem 6.32] by a generalised cellular structure 5 ([16], [12]). Moreover, unlike the situation in [11], the link between representations of Q v (n, r) F and the quantum queer supergroup has not yet been established, since we do not know if the surjective map Φ r given in Theorem 6.4 can be extended to the roots of unity case.…”

“…(2) We remark that, for the q-Schur superalgebras S v (m|n, r) F of type M with m + n ≥ r, their irreducible representations at (odd) roots of unity have been classified in [11], while a nonconstructible classification of irreducible Q v (n, r) F -supermodules is obtained in [24,Theorem 6.32] by a generalised cellular structure [12,16]. We remark that Green's codeterminant basis was a first such basis for the Schur algebra.…”

Presenting Queer Schur Superalgebras

“…The Schur superalgebras and their representations were introduced and investigated by several authors including Donkin [6] and Brundan-Kujawa [4] almost over ten years ago. Recently, the study of quantum Schur superalgebras has made substantial progress; see [17,11,12,8,9]. In particular, in [12], El Turkey and Kujawa provided a presentation of the Schur superalgebras and their quantum analogues, which generalizes the work of Doty and Giaquinto [7] for (quantum) Schur algebras.…”