A combinatorial formula for the character of the diagonal coinvariants

Abstract

Let R n be the ring of coinvariants for the diagonal action of the symmetric group S n . It is known that the character of R n as a doubly-graded S n module can be expressed using the Frobenius characteristic map as ∇e n , where e n is the n-th elementary symmetric function, and ∇ is an operator from the theory of Macdonald polynomials.We conjecture a combinatorial formula for ∇e n and prove that it has many desirable properties which support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc and Thibon. We also show that a variety of earlier conjectures and theorems on ∇e n are special cases of our conjecture.Finally, we extend our conjectures on ∇e n and several of the results supporting them to higher powers ∇ m e n .

Dual Equivalence Graphs I: A New Paradigm for Schur Positivity

“…The LLT polynomial G (k) µ (x; q), originally defined by Lascoux et al [9] in 1997, is the q-generating function of k-ribbon tableaux of shape µ weighted by a statistic called cospin. By the Stanton-White correspondence [13] weighted by a statistic called diagonal inversions is given in [6]. For a detailed account of the equivalence of these definitions (actually q a G (k) µ (x; q) = G µ (x; q) for a constant a 0 depending on µ), see [2,6].…”

“…By the Stanton-White correspondence [13] weighted by a statistic called diagonal inversions is given in [6]. For a detailed account of the equivalence of these definitions (actually q a G (k) µ (x; q) = G µ (x; q) for a constant a 0 depending on µ), see [2,6].…”

“…Since T is standard, let us abuse notation by representing a cell of T by the entry it contains. Then the set of diagonal inversions is dInv(T) = (9, 7), (9,8), (7,3), (8,3), (8,2), (3, 2), (3, 1), (2, 1), (11, 1), (11,5), (6,4), (12,4), (12,10) and so dinv(T) = 13. The diagonal descents describe the shape of T. In this case, dDes(T) = {(7, 2), (11,6), (8, 1), (9, 3), (5, 4)}.…”

“…By [6], the LLT polynomial G µ (x; q) is given by 5) where x T is the monomial x π 1 1 x π 2 2 · · · when T has weight π . Notice that when q = 1, (5.5) reduces to a product of Schur functions:…”

Parabolic Kazhdan–Lusztig polynomials for quasi-minuscule quotients

“…Given any parabolic subgroup W J in a Coxeter system (W, S), Deodhar introduced two Hecke algebra modules (one for each of the two roots q and −1 of the polynomial x 2 − (q − 1)x − q) and two families of polynomials {P J,q u,v (q)} u,v∈W J and {P J,−1 u,v (q)} u,v∈W J indexed by pairs of elements of the set of minimal coset representatives W J . These polynomials are the parabolic analogues of the Kazhdan-Lusztig polynomials: while they are related to their ordinary counterparts in several ways (see, e.g., § 2 and [7], Proposition 3.5), they also play a direct role in several areas such as the geometry of partial flag manifolds [16], the theory of Macdonald polynomials [13], [14], tilting modules [25], [26], generalized Verma modules [5], canonical bases [11], [29], the representation theory of the Lie algebra gl n [22], quantized Schur algebras [30], quantum groups [9], and physics (see, e.g., [12], and the references cited there). The computation of these polynomials is a very difficult task.…”