6 research outputs found
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
Mask formulas for cograssmannian Kazhdan-Lusztig polynomials
We give two contructions of sets of masks on cograssmannian permutations that
can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the
Iwahori-Hecke algebra. The constructions are respectively based on a formula of
Lascoux-Schutzenberger and its geometric interpretation by Zelevinsky. The
first construction relies on a basis of the Hecke algebra constructed from
principal lower order ideals in Bruhat order and a translation of this basis
into sets of masks. The second construction relies on an interpretation of
masks as cells of the Bott-Samelson resolution. These constructions give
distinct answers to a question of Deodhar.Comment: 43 page