Generalized representation stability for disks in a strip and no-k-equal spaces

Abstract

For fixed j and w, we study the jth homology of the configuration space of n labeled disks of width 1 in an infinite strip of width w. As n grows, the homology groups grow exponentially in rank, suggesting a generalized representation stability as defined by Church-Ellenberg-Farb and Ramos. We prove this generalized representation stability for the strip of width 2, leaving open the case of w > 2. We also prove it for the configuration space of n labeled points in the line, of which no k are equal.2010 Mathematics Subject Classification. 55R80 (05E10 20C30).

Representation Stability for Disks in a Strip

“…Alpert-Kahle-MacPherson proved that the rank of H k conf(n, w) grows polynomially times exponentially in n [AKM21]; as a result, the ordered configuration space of open unit-diameter disks in the infinite strip of width w is not first-order representation stable in the sense of Church-Ellenberg-Farb, Miller-Wilson. However, in the case w = 2 Alpert proved that the ordered configuration space of n open unit-diameter disks in the infinite strip of width 2 exhibits a reasonable notion of first-order representation stability, as, for all k, the homology groups H k conf(•, 2) have the structure of a finitely generated FI k+1 -module, generated in degree at most 3k, where FI k+1 is a generalization of FI# [Alp20]. One can use Alpert's results to decompose H k conf(n, 2); Q into a direct sum of irreducible S n -representations as was done in [Waw22a].…”