Affine geometric crystals in unipotent loop groups

TL;DR

This paper explores the structure of affine geometric crystals in unipotent loop groups, focusing on their products, quotients, and semi-infinite limits, linking to total positivity and representation theory.

Contribution

It constructs a quotient of affine geometric crystals within unipotent loop groups and describes its semi-infinite limit using limit ratios related to total positivity.

Findings

Constructed quotient crystals inside unipotent loop groups

Described semi-infinite limit structures of these crystals

Linked crystal structures to total positivity in loop groups

Abstract

We study products of the affine geometric crystal of type A corresponding to symmetric powers of the standard representation. The quotient of this product by the R-matrix action is constructed inside the unipotent loop group. This quotient crystal has a semi-infinite limit, where the crystal structure is described in terms of limit ratios previously appearing in the study of total positivity of loop groups.