3,330 research outputs found
Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups
We use the method of mutual interlacing to prove two conjectures on the
real-rootedness of Eulerian-like polynomials: Brenti's conjecture on
-Eulerian polynomials for Weyl groups of type , and Dilks, Petersen, and
Stembridge's conjecture on affine Eulerian polynomials for irreducible finite
Weyl groups.
For the former, we obtain a refinement of Brenti's -Eulerian polynomials
of type , and then show that these refined Eulerian polynomials satisfy
certain recurrence relation. By using the Routh--Hurwitz theory and the
recurrence relation, we prove that these polynomials form a mutually
interlacing sequence for any positive , and hence prove Brenti's conjecture.
For , our result reduces to the real-rootedness of the Eulerian
polynomials of type , which were originally conjectured by Brenti and
recently proved by Savage and Visontai.
For the latter, we introduce a family of polynomials based on Savage and
Visontai's refinement of Eulerian polynomials of type . We show that these
new polynomials satisfy the same recurrence relation as Savage and Visontai's
refined Eulerian polynomials. As a result, we get the real-rootedness of the
affine Eulerian polynomials of type . Combining the previous results for
other types, we completely prove Dilks, Petersen, and Stembridge's conjecture,
which states that, for every irreducible finite Weyl group, the affine descent
polynomial has only real zeros.Comment: 28 page
Division and the Giambelli Identity
Given two polynomials f(x) and g(x), we extend the formula expressing the
remainder in terms of the roots of these two polynomials to the case where f(x)
is a Laurent polynomial. This allows us to give new expressions of a Schur
function, which generalize the Giambelli identity.Comment: 9 pages, 1 figur
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