Stable multivariate W -Eulerian polynomials

Abstract

We prove a multivariate strengthening of Brenti's result that every root of the Eulerian polynomial of type B is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability-a generalization of realrootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator.Our results extend naturally to colored permutations, and we also give stable generalizations of recent real-rootedness results due to Dilks, Petersen, and Stembridge on affine Eulerian polynomials of types A and C. Finally, although we are not able to settle Brenti's real-rootedness conjecture for Eulerian polynomials of type D, nor prove a companion conjecture of Dilks, Petersen, and Stembridge for affine Eulerian polynomials of types B and D, we indicate some methods of attack and pose some related open problems. Conjecture 1.1 (Conjecture 5.2 in [9]). For every finite Coxeter group W, the descent generating polynomial W(x) has only real roots. Dilks, Petersen, and Stembridge later extended the definition of Eulerian polynomials to include affine descents, and proposed the following companion to Brenti's conjecture. Conjecture 1.2 (Conjecture 4.1 in [12]). For every finite Weyl group W, the affine descent generating polynomial W(x) has only real roots.Again, this was not completely proved-A n (x) and C n (x) were shown to have only real roots, the exceptional cases were verified, but the real-rootedness of the affine Eulerian polynomials of types B and D remains an open problem.

Multivariate Eulerian Polynomials and Exclusion Processes

“…In this paper we show that the multivariate partition function of the stationary distribution of the ASEP with parameters α = r/(r − 1) and β = r is a q-analogue of the multivariate Eulerian polynomials for r-coloured permutations recently introduced in [23]. This contains the signed permutations as special case for r = 2.…”

“…Recently efforts have been made to generalise Frobenius' result in yet another direction, namely to multivariate polynomials [23]. A notion of 'real-rootedness' that has been fruitful in several settings is the following.…”

“…In [23] a multivariate extension of the Eulerian polynomials for wreath products was defined in terms of descent-and ascent bottoms. By the recursion given in the proof of Theorem 3.15 in [23], we see that for a = b = 1, their polynomials are the same as ours up to a reindexing of the variables. Since F 1 = a(r − 1)x 1 + by 1 is obviously stable it remains to prove that T preserves stability.…”

“…This multivariate refinement was used in conjunction with stable polynomials in recent papers to solve the monotone column permanent conjecture [9]. Similar methods were employed to generalize properties-such as recurrences, and zero location-for several variants of Eulerian polynomials for Stirling permutations [13], barred multiset permutations [4], signed and colored permutation [23].…”

Trees

“…Then the Monotone Column Permanent Conjecture stated that the permanent of the matrix (a ij +x) n i,j=1 , where x is a variable, is real-rooted. Subsequently multivariate Eulerian polynomials for colored permutations and various other models have been studied [23,32,56,101]. is stable.…”