q-Partition algebra combinatorics |
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Authors: | Tom Halverson |
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Institution: | a Department of Mathematics, Macalester College, Saint Paul, MN 55105, United States b Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309, United States |
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Abstract: | 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 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 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 indexed by n-restricted q-set partitions of {1,…,r} and show that there are dn,r(q) of these. |
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Keywords: | Partition algebras Set partitions Finite general linear group Double centralizer RSK correspondence |
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