Schur–Weyl Reciprocity between the Quantum Superalgebra and the Iwahori–Hecke Algebra

Abstract

In this paper, we establish Schur-Weyl reciprocity between the quantum general super Lie algebra U σ q gl(m, n) and the Iwahori-Hecke algebra H Q(q),r (q). We introduce the sign q-permutation representation of H Q(q),r (q) on the tensor spaceThis action commutes with that of U σ q gl(m, n) derived from the vector representation on V. Those two subalgebras of End Q(q) (V ⊗r ) satisfy Schur-Weyl reciprocity. As special cases, we obtain the super case (q→1), and the quantum case (n = 0). Hence this result includes both the super case and the quantum case, and unifies those two important cases.

Presenting Schur superalgebras

“…It is known that the dimension of S q (m|n, d) over Q(q) equals the dimension of S(m|n, d) over Q. This is established, for example, in the proof of [19,Proposition 4.3]. This can also be seen as an outcome of [10,Theorem 9.7].…”

“…We again replace the rational numbers with the rational functions in the indeterminate q and the symmetric group by its IwahoriHecke algebra, and now replace the enveloping algebra by the quantum group associated to gl(m|n). Schur-Weyl duality in this setting was established by Moon [20] and Mitsuhashi [19]. The resulting algebra is called the q-Schur superalgebra.…”

A super Frobenius formula for the characters of Iwahori–Hecke algebras

“…In this section, we review the sign q-permutation representation of the Iwahori-Hecke algebra H q of type A ( [8]), and the vector representation of the general quantum super algebra U σ q ([1], [8]). Let R be a commutative domain with 1, and let q be an invertible element of R. The Iwahori-Hecke algebra H R,r (q) of type A is an R-algebra generated by {T i | i = 1, 2, .…”

Parastatistics Algebra, Young Tableaux and the Super Plactic Monoid

“…(quantum Schur-Weyl duality [19]) The sign permutation action π q of the Hecke algebra H r (q) and the action ρ of the quantum UEA U q (gl m|n ) on V ⊗r are centralizers to each other…”

“…The quantum version of the Schur-Weyl duality between the quantum UEA U q (gl m )-action on the tensor power of the vector representation V ⊗r and the permutation action of the Hecke algebra H r (q) is due to Jimbo [11], whereas its super-counterpart given by Theorem 5.7 was done by Mitsuhashi [19].…”