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

Brendan Pawlowski and Brendon Rhoades

Spanning line configurations

Abstract. We define and study a variety Xn,k which depends on two positive integers k <= n. When k = n, the variety Xn,k is homotopy equivalent to the flag variety Fl(n) of complete flags in Cn. We describe an affine paving of Xn,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 Sn, the geometry of Xn,k is governed by length n words on the alphabet {1,2, ..., k} in which each letter appears at least once. The space Xn,k carries a natural action of Sn, 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.

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