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. Using the quasi-symmetric function of this lattice, we can evaluate a q-analogue of the classical descent set statistic in two cases. In one case, we express the values in terms of the classical descent set statistic and find the maximal value, extending De Bruijn and Niven's results in permutation enumeration. In the other, we compute the values for certain descent sets and conjecture when the maximum is obtained. Finally, when evaluating the quasi-symmetric function using a root of unity, we obtain a version of the cyclic sieving phenomenon on the Boolean algebra, due to Reiner, Stanton and White. This talk, which is based on joint work with Richard Ehrenborg, will include abundant examples and be accessible to anyone who has a minimal knowledge of combinatorics.