2024/11/22 by Du, Jie, Gu, Haixia, Li, Zhenhua +1 · 1 citation
#17A70 #17B37 #20C08 #20G42 #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2411.14764
In [DGLW], we use certain special elements and their commutation relations in the Hecke-Clifford algebras Hcr,R to derive some fundamental multiplication formulas associated with the natural bases in queer q-Schur superalgebras Qq(n,r;R) introduced in [DW2]. Here a natural basis element is defined by a special element TA⋆ in Hcr,R associated with a pair of certain n× n matrices A⋆=(A0|A1) over ℕ with entries sum to r. The definition of TA^⋆ consists of an element cA⋆ in the Clifford superalgebra and an element TA in the Hecke algebra, where A=A0+A1. Note that all TA can be used to define the natural basis for the corresponding q-Schur algebra Sq(n,r). This paper is a continuation of [DGLW]. We start with standardized queer v-Schur superalgebras Qsv(n,r), for R=ℤ[v,v-1] and q=v2, and their natural bases. With the v-Schur algebra Sv(n,r) at the background, the first key ingredient is a standardisation of the natural basis for Qsv(n,r) and their associated standard multiplication formulas. By introducing some long elements of finite sums, we then extend the formulas to these long elements which allow us to explicitly define ℚ(v)-superalgebra homomorphisms ξn,r from the quantum queer supergroup \boldsymbolUv(\mathfrakqn) to queer q-Schur superalgebras \boldsymbolQsv(n,r), for all r≥1. Finally, taking limits of long elements yields certain infinitely long elements as formal infinite series which eventually lead to a new construction for \boldsymbolUv(\mathfrakqn).