Séminaire Lotharingien de Combinatoire, 82B.15 (2019), 12 pp.

Brendan Pawlowski and Brendon Rhoades

Spanning line configurations

Abstract. We define and study a variety X_n,k which depends on two positive integers k <= n. When k = n, the variety X_n,k is homotopy equivalent to the flag variety Fl(n) of complete flags in Cⁿ. We describe an affine paving of X_n,k, present its cohomology, and describe the cellular cohomology classes in terms of Schubert polynomials. Just as the geometry of Fl(n) is governed by the combinatorics of permutations in S_n, the geometry of X_n,k is governed by length n words on the alphabet {1,2, ..., k} in which each letter appears at least once. The space X_n,k carries a natural action of S_n, and we relate the induced cohomology representation to Macdonald theory via the Delta Conjecture of Haglund, Remmel, and Wilson.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

The following versions are available:

PDF (191 K)
TeX version