q-Partition algebra combinatorics

We study a q-analog Q r ( n , q ) of the partition algebra P r ( n ) . The algebra Q r ( n , q ) arises as the centralizer algebra of the finite general linear group GL n ( F q ) acting on a vector space IR q r coming from r-iterations of Harish–Chandra restriction and induction. For n ⩾ 2 r , we show that Q r ( n , q ) has the same semisimple matrix structure as P r ( n ) . We compute the dimension d n , r ( q ) = dim ( IR q r ) to be a q-polynomial that specializes as d n , r ( 1 ) = n r and d n , r ( 0 ) = B ( r ) , the rth Bell number. Our method is to write d n , r ( q ) as a sum over integer sequences which are q-weighted by inverse major index. We then find a basis of IR q r indexed by n-restricted q-set partitions of { 1 , … , r } and show that there are d n , r ( q ) of these.