Polynomials associated to Lie algebras

2025/06/30 by Bruna, Matías, Capuñay, Alex, Friedman, Eduardo
#11M41 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2507.00326

Abstract

We associate to a semisimple complex Lie algebra \mathfrakg a sequence of polynomials P_ℓ,\mathfrakg(x)∈ℚ[x] in r variables, where r is the rank of \mathfrakg and ℓ=0,1,2,… . The polynomials P_ℓ,\mathfrakg(x) are uniquely associated to the isomorphism class of \mathfrakg, up to re-numbering the variables, and are defined as special values of a variant of Witten's zeta function. Another set of polynomials associated to \mathfrakg were defined in 2008 by Komori, Matsumoto and Tsumura using different special values of another variant of Witten's zeta function.

Citations

• Vanishing of Witten zeta function at negative integers
• Zeros of Witten zeta functions and absolute limit
• Special values of Dirichlet series and zeta integrals

Related

• Pro-isomorphic zeta functions of some D^∗ Lie lattices of even rank
• Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials
• Theta and eta polynomials in geometry, Lie theory, and combinatorics
• More on characteristic polynomials of Lie algebras
• The \mathfraksl₂-actions on the symmetric polynomials and on Young diagrams