##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics Seminar

## Gidon Orelowitz

#### UIUC

## The Kostka semigroup and its Hilbert basis

##### Abstract:

The Kostka semigroup consists of pairs of partitions with at most r parts that have a positive Kostka coefficient. For this semigroup, Hilbert basis membership is an NP-complete problem. We introduce KGR graphs and conservative subtrees, through the Gale-Ryser theorem on contingency tables, as a criterion for membership. In our main application, we show that if a partition pair is in the Hilbert basis then the partitions are at most $r$ wide. We also classify the extremal rays of the associated polyhedral cone; these rays correspond to a (strict) subset of the Hilbert basis. In an appendix, the second and third authors show that a natural extension of our main result on the Kostka semigroup cannot be extended to the Littlewood-Richardson semigroup. This furthermore gives a counterexample to recent speculation of P. Belkale concerning the semigroup controlling nonvanishing conformal blocks.

Host: Brendon Rhoades

### November 15, 2022

### 4:00 PM

APM 5829

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