q-Partition algebra combinatorics

We study a q-analog Qr(n, q) of the partition algebra Pr(n). The algebra Qr(n, q) arises as the centralizer algebra of the finite general linear group GLn(Fq) acting on a vector space IR q coming from r-iterations of Harish-Chandra restriction and induction. For n ≥ 2r, we show that Qr(n, q) has the same semisimple matrix structure as Pr(n). We compute the dimension dn,r(q) = dim(IR r q ) to be a q-polynomial that specializes as dn,r(1) = n r and dn,r(0) = B(r), the rth Bell number. Our method is to write dn,r(q) as a sum over integer sequences which are q-weighted by inverse major index. We then find a basis of IRr q indexed by n-restricted q-set partitions of {1, . . . , r} and show that there are dn,r(q) of these. Introduction The general linear group GLn(C) and the symmetric group Sr both act on tensor space V ⊗r, where V is the natural n dimensional representation of GLn(C) and Sr acts by tensor place permutations. Classical Schur–Weyl duality says that these actions commute and that each action generates the full centralizer of the other, so that as a (GLn(C), Sr)-bimodule the tensor space has a multiplicity-free decomposition given by V ⊗r ∼= ⊕ λ L(λ)⊗ S λ r , where the L(λ) are irreducible GLn(C)-modules and the S λ r are the irreducible Sr-modules. If we restrict GLn(C) to its subgroup of orthogonal matrices On(C), then the centralizer algebra is Brauer’s centralizer algebra Br(n) = EndOn(C)(V ⊗r). If we restrict further to the symmetric groups Sn−1 ⊆ Sn ⊆ On(C) ⊆ GLr(C), then the centralizer algebras are the partition algebras Pr(n) = EndSn(V ⊗r) and Pr+ 1 2 (n) = EndSn−1(V ⊗r). Furthermore, the containments reverse: subgroup G ⊆ GLn(C) : Sn−1 ⊆ Sn ⊆ On(C) ⊆ GLn(C) l l l centralizer algebra EndG(V ⊗r) : Pr+ 1 2 (n) ⊇ Pr(n) ⊇ Br(n) ⊇ CSr. The Brauer algebras were introduced in 1937 by Richard Brauer. The partition algebras arose early in the 1990s in the work of Martin [Mar1], [Mar2] and later, independently, in the work of Jones [Jo] (see also [HR]). For r ∈ 1 2Z>0, the partition algebra Pr(n) has a basis indexed by the set partitions of {1, 2, . . . , 2r} and a multiplication given by “diagram multiplication.”