# ALGEBRAIC COMBINATORICS

A Demazure Character Formula for the Product Monomial Crystal
Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 301-327.

The product monomial crystal was defined by Kamnitzer, Tingley, Webster, Weekes, and Yacobi for any semisimple simply-laced Lie algebra $𝔤$, and depends on a collection of parameters $\mathbf{R}$. We show that a family of truncations of this crystal are Demazure crystals, and give a Demazure-type formula for the character of each truncation, and the crystal itself. This character formula shows that the product monomial crystal is the crystal of a generalised Demazure module, as defined by Lakshmibai, Littelmann and Magyar. In type $A$, we show the product monomial crystal is the crystal of a generalised Schur module associated to a column-convex diagram depending on $\mathbf{R}$.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.156
Classification: 05E10
Keywords: Monomial crystal, generalised schur module, Demazure crystal.
Gibson, Joel 1

1 School of Mathematics and Statistics University of Sydney NSW 2006, Australia
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Gibson, Joel. A Demazure Character Formula for the Product Monomial Crystal. Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 301-327. doi : 10.5802/alco.156. https://alco.centre-mersenne.org/articles/10.5802/alco.156/

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