第96回　2019年9月9日 14:00〜15:30

石井 基裕（群馬大学）「Level-zero crystals of semi-infinite Young tableaux of untwisted affine type $A$」

It is known that every fundamental representation of the special linear Lie algebra $\mathfrak{sl}_n$ is a minuscule representation and, in consequence, a (crystal) basis of any finite-dimensional representation of $\mathfrak{sl}_n$ is parametrized by (semi-standard) Young tableaux. Similarly, it follows that any "fundamental" extremal weight module over the quantized universal enveloping algebra of untwisted affine type $A$ is a minuscule representation. Motivated by this fact, I introduce the crystal of "semi-infinite Young tableaux," which gives a new combinatorial model for the crystal basis of any level-zero extremal weight module over the quantized universal enveloping algebra of untwisted affine type $A$.

I will talk about
(1) crystal bases of extremal weight modules over quantized universal enveloping algebras of (untwisted) affine types, following [Kashiwara (2002)] and [Beck-Nakajima (2004)],
(2) a characterization of such crystal bases in terms of semi-infinite Bruhat order on affine Weyl groups via semi-infinite Lakshmibai-Seshadri paths,
(3) the definition of semi-infinite Young tableaux,
(4) a tableau criterion for the semi-infinite Bruhat order on affine Weyl groups of type $A$,
(5) a characterization of semi-infinite Young tableaux, and
(6) an explicit description of the crystal structure on the set of semi-infinite Young tableaux.