Linear idempotents in Matsuo algebras

TL;DR

This paper studies idempotents in Matsuo algebras, demonstrating their Seress property, calculating eigenvalues for symmetric groups, and relating involutions in D_n root systems to algebraic idempotents.

Contribution

It establishes the Seress property for a broad class of idempotents, generalizes eigenvalue calculations, and links Weyl group involutions to algebraic idempotents.

Findings

Idempotents satisfy the Seress property, ensuring well-behaved algebraic structure.

Eigenvalues in Matsuo algebras of symmetric groups are computed for any parameter.

In D_n, conjugacy classes of involutions correspond to idempotents via fusion rules.

Abstract

Matsuo algebras are an algebraic incarnation of 3-transposition groups with a parameter $\alpha$, where idempotents takes the role of the transpositions. We show that a large class of idempotents in Matsuo algebras satisfy the Seress property, making these nonassociative algebras well-behaved analogously to associative algebras, Jordan algebras and vertex (operator) algebras. We calculate eigenvalues in the Matsuo algebra of ${\rm Sym}(n)$ for any $\alpha$, generalising some vertex algebra results for which $\alpha=\frac{1}{4}$. Finally, in the Matsuo algebra of the root system ${\rm D}_n$, we show $n-3$ conjugacy classes of involutions coming from the Weyl group are in natural bijection with idempotents in the algebra via their fusion rules.