58 research outputs found
From Bruhat intervals to intersection lattices and a conjecture of Postnikov
We prove the conjecture of A. Postnikov that (A) the number of regions in the
inversion hyperplane arrangement associated with a permutation w\in \Sn is at
most the number of elements below in the Bruhat order, and (B) that
equality holds if and only if avoids the patterns 4231, 35142, 42513 and
351624. Furthermore, assertion (A) is extended to all finite reflection groups.
A byproduct of this result and its proof is a set of inequalities relating
Betti numbers of complexified inversion arrangements to Betti numbers of closed
Schubert cells. Another consequence is a simple combinatorial interpretation of
the chromatic polynomial of the inversion graph of a permutation which avoids
the above patterns.Comment: 24 page
From Bruhat intervals to intersection lattices and a conjecture of Postnikov
We prove the conjecture of A. Postnikov that () the number of regions in the inversion hyperplane arrangement associated with a permutation is at most the number of elements below in the Bruhat order, and () that equality holds if and only if avoids the patterns , , and . Furthermore, assertion () is extended to all finite reflection groups
One-skeleton posets of Bruhat interval polytopes
Introduced by Kodama and Williams, Bruhat interval polytopes are generalized
permutohedra closely connected to the study of torus orbit closures and total
positivity in Schubert varieties. We show that the 1-skeleton posets of these
polytopes are lattices and classify when the polytopes are simple, thereby
resolving open problems and conjectures of Fraser, of Lee--Masuda, and of
Lee--Masuda--Park. In particular, we classify when generic torus orbit closures
in Schubert varieties are smooth.Comment: 19 page
Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams
We consider the problem of finding the number of matrices over a finite field
with a certain rank and with support that avoids a subset of the entries. These
matrices are a q-analogue of permutations with restricted positions (i.e., rook
placements). For general sets of entries these numbers of matrices are not
polynomials in q (Stembridge 98); however, when the set of entries is a Young
diagram, the numbers, up to a power of q-1, are polynomials with nonnegative
coefficients (Haglund 98).
In this paper, we give a number of conditions under which these numbers are
polynomials in q, or even polynomials with nonnegative integer coefficients. We
extend Haglund's result to complements of skew Young diagrams, and we apply
this result to the case when the set of entries is the Rothe diagram of a
permutation. In particular, we give a necessary and sufficient condition on the
permutation for its Rothe diagram to be the complement of a skew Young diagram
up to rearrangement of rows and columns. We end by giving conjectures
connecting invertible matrices whose support avoids a Rothe diagram and
Poincar\'e polynomials of the strong Bruhat order.Comment: 24 pages, 9 figures, 1 tabl
Combinatorics in Schubert varieties and Specht modules
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, June 2011."June 2011." Cataloged from PDF version of thesis.Includes bibliographical references (p. 57-59).This thesis consists of two parts. Both parts are devoted to finding links between geometric/algebraic objects and combinatorial objects. In the first part of the thesis, we link Schubert varieties in the full flag variety with hyperplane arrangements. Schubert varieties are parameterized by elements of the Weyl group. For each element of the Weyl group, we construct certain hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial if and only if the Schubert variety is rationally smooth. For classical types the arrangements are (signed) graphical arrangements coning from (signed) graphs. Using this description, we also find an explicit combinatorial formula for the Poincaré polynomial in type A. The second part is about Specht modules of general diagram. For each diagram, we define a new class of polytopes and conjecture that the normalized volume of the polytope coincides with the dimension of the corresponding Specht module in many cases. We give evidences to this conjecture including the proofs for skew partition shapes and forests, as well as the normalized volume of the polytope for the toric staircase diagrams. We also define new class of toric tableaux of certain shapes, and conjecture the generating function of the tableaux is the Frobenius character of the corresponding Specht module. For a toric ribbon diagram, this is consistent with the previous conjecture. We also show that our conjecture is intimately related to Postnikov's conjecture on toric Specht modules and McNamara's conjecture of cylindric Schur positivity.by Hwanchul Yoo.Ph.D
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