# Gelfand-Tsetlin polytopes: a story of flow and order polytopes

## TL;DR

This paper explores the deep connections between Gelfand-Tsetlin polytopes, order polytopes, and flow polytopes, revealing new theoretical insights in algebraic combinatorics and representation theory.

## Contribution

It establishes a general theory linking marked order polytopes to flow polytopes, building on the recent identification of Gelfand-Tsetlin polytopes as flow polytopes.

## Key findings

- Gelfand-Tsetlin polytopes are marked order polytopes and flow polytopes.
- Derived corollaries connecting combinatorial and geometric properties.
- Provided a unified framework for understanding these polytopes.

## Abstract

Gelfand-Tsetlin polytopes are prominent objects in algebraic combinatorics. The number of integer points of the Gelfand-Tsetlin polytope $\mathrm{GT}(\lambda)$ is equal to the dimension of the corresponding irreducible representation of $GL(n)$. It is well-known that the Gelfand-Tsetlin polytope is a marked order polytope; the authors have recently shown it to be a flow polytope. In this paper, we draw corollaries from this result and establish a general theory connecting marked order polytopes and flow polytopes.