Crystal Invariant Theory

Abstract: Abstract: Berenstein and Kazhdan have introduced a birational lifting of Kashiwara's crystals, called geometric crystals. Their theory gives rise to four families of operators acting on the space of complex $m \times n$ matrices, two acting by geometric crystal operators and two acting by geometric R-matrices. These actions can be viewed as "crystal analogues" of the usual actions of $GL_m$ and $GL_n$ - and their subgroups $S_m$ and $S_n$ - on the polynomial ring in $m \times n$ variables. Many important functions in the theory of geometric crystals are invariants of one or more of those actions. The examples include $\epsilon$ and $\phi$ functions, energy function, decoration function, insertion and recording tableaux of Noumi-Yamada geometric RSK, central charge, etc. We study generators of the invariants of one or more of the families of operators, and obtain new formulas for the important functions by writing them in terms of those generators. The talk is based on joint work with Ben Brubaker, Gabe Frieden, Travis Scrimshaw, and Thomas Lam.

commutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Comments: Zoom meeting ID: 927 9300 3904 Password: 909921

SJTU algebra seminar