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Register a new userA combinatorial formula for the nabla operator
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We present an LLT-type formula for a general power of the nabla operator applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalized shuffle theorem describing $ abla^k e_n$, and the Elias-Hogancamp formula for $( abla^k p_1^n,e_n)$ as corollaries. We give a direct proof of the theorem by verifying that the LLT expansion satisfies the defining properties of $ abla^k$, such as triangularity in the dominance order, as well as a geometric proof based on a method for counting bundles on $mathbb{P}^1$ due to the second author. These formulas are related to an affine paving of the type A unramified affine Springer fiber studied by Goresky, Kottwitz, and MacPherson, and also to Stanleys chromatic symmetric functions.
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The operator nabla, introduced by Garsia and the author, plays a crucial role in many aspect of the study of diagonal harmonics. Besides giving several new formulas involving this operator, we show how one is lead to representation theoretic explanations for conjectures about the effect of this operator on Schur functions.
We present a decomposition of the generalized binomial coefficients associated with Jack polynomials into two factors: a stem, which is described explicitly in terms of hooks of the indexing partitions, and a leaf, which inherits various recurrence properties from the binomial coefficients and depends exclusively on the skew diagram. We then derive a direct combinatorial formula for the leaf in the special case where the two indexing partitions differ by at most two rows. This formula also exhibits an unexpected symmetry with respect to the lengths of the two rows.
We revisit the geometric description of cluster categories in type E in terms of colored diagonals in a polygon and generalize it to the case of m-cluster categories. As an application, we relate colored diagonals in a polygon to semi-standard Young tableaux, in type E_6,E_7,E_8. This provides a new compatibility description of semi--standard Young tableaux in Grassmannian cluster algebras in type E_6, E_8 and in a sub-cluster algebra of type E_7.
The multiplicity of a weight in a finite-dimensional irreducible representation of a simple Lie algebra g can be computed via Kostants weight multiplicity formula. This formula consists of an alternating sum over the Weyl group (a finite group) and involves a partition function known as Kostants partition function. Motivated by the observation that, in practice, most terms in the sum are zero, our main results describe the elements of the Weyl alternation sets. The Weyl alternation sets are subsets of the Weyl group which contributes nontrivially to the multiplicity of a weight in a highest weight representation of the Lie algebras so_4(C), so_5(C), sp_4(C), and the exceptional Lie algebra g_2. By taking a geometric approach, we extend the work of Harris, Lescinsky, and Mabie on sl_3(C), to provide visualizations of these Weyl alternation sets for all pairs of integral weights lambda and mu of the Lie algebras considered.
The $q$-analog of Kostants weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the $q$-analog of Kostants partition function. This formula, when evaluated at $q=1$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $mathfrak{sl}_4(mathbb{C})$ and give closed formulas for the $q$-analog of Kostants weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the $q$-analog of Kostants partition function by counting restricted colored integer partitions. These formulas, when evaluated at $q=1$, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostants weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of $mathfrak{sl}_4(mathbb{C})$, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse.
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