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Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements

Cédric Lecouvey, David Wahiche

TL;DR

The paper develops a unifying framework connecting the Nekrasov–Okounkov hook-length formulas with the Weyl–Kac denominator for all classical affine root systems by reexpressing renormalized denominators as sums over affine Grassmannian elements. It achieves this through foldings of Dynkin diagrams, weighted box models on self-conjugate cores, and Littlewood decomposition, linking atomic length to partition statistics and to hook-length products. The approach yields explicit atomic-length formulas across types (A, B, C, D, G, and twisted variants) and provides a bridge between affine Grassmannian elements and standard partition combinatorics via($V_{g,n}$-coding). These results generalize Nekrasov–Okounkov-type identities beyond type A, offering a systematic, combinatorial mechanism to derive hook-length expressions from affine-root data, with potential implications for representation theory and partition identities.

Abstract

The Nekrasov-Okounkov formula gives an expression for the Fourier coefficients of the Euler functions as a sum of hook length products. This formula can be deduced from a specialization in a renormalization of the affine type $A$ Weyl denominator formula and the use of a polynomial argument. In this paper, we rephrase the renormalized Weyl-Kac denominator formula as a sum parametrized by affine Grassmannian elements. This naturally gives rise to the (dual) atomic length of the root system considered introduced by Chapelier-Laget and Gerber. We then provide an interpretation of this atomic length as the cardinality of some subsets of $n$-core partitions by using foldings of affine Dynkin diagrams. This interpretation does not permit the direct use of a polynomial argument for all affine root systems. We show that this obstruction can be overcome by computing the atomic length of certain families of integer partitions. Then we show how hook-length statistics on these partitions are connected with the Coxeter length on affine Grassmannian elements and Nekrasov-Okounkov type formulas.

Macdonald Identities, Weyl-Kac Denominator Formulas and Affine Grassmannian Elements

TL;DR

The paper develops a unifying framework connecting the Nekrasov–Okounkov hook-length formulas with the Weyl–Kac denominator for all classical affine root systems by reexpressing renormalized denominators as sums over affine Grassmannian elements. It achieves this through foldings of Dynkin diagrams, weighted box models on self-conjugate cores, and Littlewood decomposition, linking atomic length to partition statistics and to hook-length products. The approach yields explicit atomic-length formulas across types (A, B, C, D, G, and twisted variants) and provides a bridge between affine Grassmannian elements and standard partition combinatorics via(-coding). These results generalize Nekrasov–Okounkov-type identities beyond type A, offering a systematic, combinatorial mechanism to derive hook-length expressions from affine-root data, with potential implications for representation theory and partition identities.

Abstract

The Nekrasov-Okounkov formula gives an expression for the Fourier coefficients of the Euler functions as a sum of hook length products. This formula can be deduced from a specialization in a renormalization of the affine type Weyl denominator formula and the use of a polynomial argument. In this paper, we rephrase the renormalized Weyl-Kac denominator formula as a sum parametrized by affine Grassmannian elements. This naturally gives rise to the (dual) atomic length of the root system considered introduced by Chapelier-Laget and Gerber. We then provide an interpretation of this atomic length as the cardinality of some subsets of -core partitions by using foldings of affine Dynkin diagrams. This interpretation does not permit the direct use of a polynomial argument for all affine root systems. We show that this obstruction can be overcome by computing the atomic length of certain families of integer partitions. Then we show how hook-length statistics on these partitions are connected with the Coxeter length on affine Grassmannian elements and Nekrasov-Okounkov type formulas.
Paper Structure (32 sections, 36 theorems, 201 equations, 7 figures, 5 tables)

This paper contains 32 sections, 36 theorems, 201 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Affine root systems and affine Lie algebras
  3. Macdonald formula types
  4. Connection with the affine Grassmannian
  5. Atomic length and cores
  6. Affine type A and core partitions
  7. Type $\boldsymbol{C_{n}^{(1)}}$
  8. Type $\boldsymbol{D_{n+1}^{(2)}}$
  9. Type $\boldsymbol{A_{2n}^{(2)}}$
  10. Type $\boldsymbol{A_{2n}^{\prime (2)}}$
  11. Type $\boldsymbol{B_{n}^{(1)}}$
  12. Type $\boldsymbol{A_{2n-1}^{(2)}}$
  13. Type $\boldsymbol{D_{n}^{(1)}}$
  14. Weighted boxes in cores
  15. Type $\boldsymbol{D_{4}^{(3)}}$
  16. ...and 17 more sections

Key Result

Lemma 2.1

For any $j\in I^{\ast}$, we have that is, the weights $\omega_{i}$, $i\in I^{\ast}$ and the roots $\alpha_{i}$, $i\in I^{\ast}$ can be regarded as the dominant weights and the simple roots of the finite root system with Cartan matrix $(a_{i,j})_{(i,j)\in I^{\ast}\times I^{\ast}}$.

Figures (7)

  • Figure 1: The Dynkin diagrams of the extended simple root systems.
  • Figure 2: The Dynkin diagrams of the twisted simple root systems.
  • Figure 3: The Dynkin diagram of type $C_n^{(1)}$ by gluing nodes of the Dynkin diagram of type $A_{2n-1}^{(1)}$.
  • Figure 4: $\partial\lambda$ and its binary correspondence for $\lambda=(5,5,3,2)$ with a hook.
  • Figure 5: Distinct partition, a self-conjugate partition, a doubled distinct partition and its conjugate filled with $\varepsilon$: (a) shifted Young diagram of $\bar{\lambda}=(5,2,1)\in\mathcal{D}$, (b) $\lambda=(5,3,3,1,1)\in\mathcal{SC}$, (c) $\lambda=(6,4,4,1,1)\in\mathcal{DD}$, (d) $\lambda=(5,3,3,3,1,1)\in\mathcal{DD}^{\mathrm{tr}}$.
  • ...and 2 more figures

Theorems & Definitions (57)

  • Lemma 2.1
  • Remark 3.1
  • Lemma 4.1
  • Remark 4.2
  • Theorem 4.3
  • Lemma 4.4
  • Remark 4.5
  • Example 5.1
  • Proposition 5.2
  • Lemma 5.3
  • ...and 47 more