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Equivariant Kazhdan-Lusztig polynomials of \(q\)-niform matroids. (English) Zbl 1417.05024
Summary: We study \(q\)-analogues of uniform matroids, which we call \(q\)-niform matroids. While uniform matroids admit actions of symmetric groups, \(q\)-niform matroids admit actions of finite general linear groups. We show that the equivariant Kazhdan-Lusztig polynomial of a \(q\)-niform matroid is the unipotent \(q\)-analogue of the equivariant Kazhdan-Lusztig polynomial of the corresponding uniform matroid, thus providing evidence for the positivity conjecture for equivariant Kazhdan-Lusztig polynomials.

MSC:
05B35 Combinatorial aspects of matroids and geometric lattices
05C31 Graph polynomials
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
20C33 Representations of finite groups of Lie type
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