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NCUR Conference Talk

Date:
-
Location:
NCUR Conference
Speaker(s) / Presenter(s):
Cyrus Hettle, University of Kentucky

Title:  On the flag enumeration of the subspace lattice

Abstract:   We consider the q-analogue of the Boolean algebra: the lattice of subspaces of an n-dimensional vector space over the finite field of q elements. The quasi-symmetric function (Ehrenborg, 1996) of this lattice encodes the flag f-vector which is the multinomial Gaussian coefficients. We express the quasi-symmetric function using quasi-symmetric functions in variables that q-commute, which can be seen as an extension of the q-binomial theorem. All the expressions of q are in fact polynomials in this variable. Hence it is natural to evaluate these expressions for other values than prime powers. When setting q to be a root of unity we obtain a version of the cyclic sieving phenomenon (due to Reiner, Stanton and White, 2004) of the Boolean algebra on the subspace lattice. Finally, we present applications of the ab-index on the subspace lattice to a descent set statistic on permutations, which is a q-analogue of the classical descent set statistic evaluated using the inversion statistics of the permutations with given descent set. When we evaluate the flag h-vector of the subspace at a primitive kth root of unity, all of the values are real when k divides n or n-1. Furthermore, when k divides n we are able to determine the extreme values, extending results of De Bruijn (1970) and Niven (1968). When k divides n-1 we are able to express certain entries of the vector in terms of the descent set statistic.