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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 **C**^{n}. 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.

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