Abstract
We introduce a series of conjectured identities that deform Weyl's denominator formula and generalize Tokuyama's formula to other root systems. These conjectures generalize a number of well-known results due to Okada. We also prove a related result for $B'_n$ that generalizes a theorem of Simpson. Nous proposons une série de conjectures qui sont des déformations de la formule dénominateur de Weyl et qui généralisent la formule de Tokuyama à d’autres systèmes de racines. Ces résultats sont des généralisations de théorèmes bien connus dus à Okada. Nous donnons aussi la preuve d’un résultat pour $B'_n$ qui est une généralisation d’un théorème de Simpson.
Highlights
Weyl’s denominator formula is a fundamental and well-studied identity in algebra; it is valued for its connections to combinatorics and analytic number theory
Deformations of Weyl’s denominator formula for the root systems of type Bn, Cn and Dn have been discovered by Okada [11], and for Bn a further deformation was obtained by Simpson [13]
The setting for all these identities is a particular type of alternating sign matrix, described either as an ASM invariant under 180 degree rotation (Okada [11]) or as a half-turn ASM (Robbins [12], Kuperberg [9], Tabony [15])
Summary
Weyl’s denominator formula is a fundamental and well-studied identity in algebra; it is valued for its connections to combinatorics and analytic number theory. From this query we first developed a number of conjectures for cases that we refer to as Bn, Bn, Cn, Cn, Dn, and Dn (the six conjectures) that generalise to a multi-parameter setting the one-parameter results of Okada and Simpson, and have proved a version of Tokuyama’s identity for Bn (the one result). This more general result whose proof we discuss here encompasses the three Bn, Dn and Bn conjectures that are themselves generalisations of Okada’s Theorems 2.1 and 4.1 [11] and Simpson’s Theorem 1 [13]. 3 introduces the conjectures, and, Section 4 gives the result and specializations
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