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 IRr 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(IRr q) to be a q-polynomial that specializes as dn,r(1) = nr 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.

• Type: Journal
• Year: 2010
• Where: JCT
• Authors: Tom Halverson, Nathaniel Thiem
• Keywords: Algebra, JCT 2010, Vector Space Irr, finite General Linear