representation theory

Cluster structures on $S L_{n} / S O_{n}$

arXiv:2607.14634

summary

The paper constructs cluster algebra structures on the strata of the symmetric space SL_n/SO_n by folding upper cluster algebras from SL_n, and shows these structures are compatible with De Concini Poisson brackets while also providing cluster structures for symmetric matrices.

Abstract

We construct cluster structures on the strata $S ˚_{w}$ of a stratification of the symmetric space $S L_{n} / S O_{n}$ over $C$ . To accomplish this, we study foldings of upper cluster algebras and show that the cluster structures on $S L_{n} / S O_{n}$ can be obtained from those on $S L_{n}$ via folding. As a corollary, we show that these cluster structures are compatible with the De Concini Poisson structures, and we construct cluster structures on the variety of symmetric matrices $Sym_{n}$ .

21 pages

Topics & keywords

#cluster algebras #symmetric spaces #folding #poisson geometry #algebraic groupsSL_nSO_nupper cluster algebraDe Concini Poisson structureSym_nfolding

Cluster structures on SLn​/SOn​

Abstract

Topics & keywords

Cluster structures on $S L_{n} / S O_{n}$