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Kostant $ρ$-decomposition of homology I. Finite-dimensional representations

Steven V Sam, Keller VandeBogert, Jerzy Weyman

TL;DR

This work develops explicit, uniform formulas for the graded characters and total ranks of Lie algebra homology for finite-dimensional representations across all classical types, revealing divisibility by a large power of 1+t and an equidistribution structure. The authors introduce a Generalized ρ-Decomposition, relating total homology characters to products of top/bottom Weyl-characters with a 2^n factor, via distinguished parabolic decompositions; they provide determinant-based combinatorial identities that control these decompositions in types A,B,C,D. The second half reinterprets these identities algebraically through Kostant’s theorem, deriving closed-form expressions for the total characters and dimensions, including specialized results for exterior algebras and the standard A-type case that connects to Eisenbud–Fløystad–Weyman free resolutions. Across all types, the authors establish a pervasive pattern: graded homology characters are divisible by (1+t)^n, admit 2^n regroupings into identical blocks, and yield explicit product formulas in terms of top and bottom Weyl-data. These results illuminate deep links between representation theory, combinatorics of Weyl groups, and homological rank conjectures, and have potential applications to Tor computations over polynomial rings and to equidistribution phenomena in syzygies and nilpotent orbit theory.

Abstract

We give explicit, uniform formulas for the graded characters and total ranks of the Lie algebra homology of finite-dimensional representations in all classical types. In many cases, these compute the Tor groups of finite length modules over polynomial rings, and this is the first in a series of papers to investigate total rank conjectures from this perspective. These formulas refine and generalize the classical $ρ$-decomposition of Kostant, and in particular we prove that the characters involved exhibit three structural phenomena: divisibility (by a large power of 2), equidistribution, and uniform factorization formulas.

Kostant $ρ$-decomposition of homology I. Finite-dimensional representations

TL;DR

This work develops explicit, uniform formulas for the graded characters and total ranks of Lie algebra homology for finite-dimensional representations across all classical types, revealing divisibility by a large power of 1+t and an equidistribution structure. The authors introduce a Generalized ρ-Decomposition, relating total homology characters to products of top/bottom Weyl-characters with a 2^n factor, via distinguished parabolic decompositions; they provide determinant-based combinatorial identities that control these decompositions in types A,B,C,D. The second half reinterprets these identities algebraically through Kostant’s theorem, deriving closed-form expressions for the total characters and dimensions, including specialized results for exterior algebras and the standard A-type case that connects to Eisenbud–Fløystad–Weyman free resolutions. Across all types, the authors establish a pervasive pattern: graded homology characters are divisible by (1+t)^n, admit 2^n regroupings into identical blocks, and yield explicit product formulas in terms of top and bottom Weyl-data. These results illuminate deep links between representation theory, combinatorics of Weyl groups, and homological rank conjectures, and have potential applications to Tor computations over polynomial rings and to equidistribution phenomena in syzygies and nilpotent orbit theory.

Abstract

We give explicit, uniform formulas for the graded characters and total ranks of the Lie algebra homology of finite-dimensional representations in all classical types. In many cases, these compute the Tor groups of finite length modules over polynomial rings, and this is the first in a series of papers to investigate total rank conjectures from this perspective. These formulas refine and generalize the classical -decomposition of Kostant, and in particular we prove that the characters involved exhibit three structural phenomena: divisibility (by a large power of 2), equidistribution, and uniform factorization formulas.

Paper Structure

This paper contains 38 sections, 30 theorems, 166 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. A Generalized $\rho$-Decomposition
  3. Organization
  4. Combinatorial Aspects
  5. Preliminaries
  6. Fixed notation
  7. Characters
  8. Identities
  9. Type A Combinatorial Identities
  10. Setup
  11. Determinantal Form
  12. Equidistribution
  13. Factorization
  14. Type BCD Combinatorial Identities
  15. Setup
  16. ...and 23 more sections

Key Result

Theorem 1.1

Set $m = 2n + \varepsilon$ where $\varepsilon \in \{ 0 ,1 \}$. Let $\mathfrak{g}$ be classical of type $\mathrm{A,B,C,D}$ with a parabolic subalgebra $\mathfrak{p}=\mathfrak{l}\oplus\mathfrak{n}$ as in Table tab:parabolics, and let $L^\mathfrak{g}_\lambda$ be the finite-dimensional irreducible of hi

Figures (1)

  • Figure 1: Left: Hasse diagram for extended Gale order on $\mathcal{P}([3])$. Right: Hasse diagram for semi-extended Gale order on $\mathcal{P}([4])$.

Theorems & Definitions (85)

  • Theorem 1.1: Generalized $\rho$-decomposition
  • Corollary 1.2
  • Example 2.2
  • Proposition 2.4: Weyl character formulas: determinantal forms
  • Remark 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • proof
  • ...and 75 more