# Newton-Okounkov polytopes of flag varieties for classical groups

## TL;DR

This paper constructs and analyzes Newton-Okounkov polytopes for classical groups' flag varieties, linking them to known polytopes in types A and C and exploring new examples in types B and D.

## Contribution

It introduces uniform geometric valuations for classical groups' flag varieties and identifies their Newton-Okounkov polytopes with known and new polytopes across different types.

## Key findings

- Identified Newton-Okounkov polytopes with Feigin-Fourier-Littelmann-Vinberg polytopes in types A and C.
- Computed low-dimensional examples for types B and D.
- Formulated open questions for further research.

## Abstract

For classical groups SL(n), SO(n) and Sp(2n), we define uniformly geometric valuations on the corresponding complete flag varieties. The valuation in every type comes from a natural coordinate system on the open Schubert cell and is combinatorially related to the Gelfand-Zetlin pattern in the same type. In types A and C, we identify the corresponding Newton-Okounkov polytopes with the Feigin-Fourier-Littelmann-Vinberg polytopes. In types B and D, we compute low-dimensional examples and formulate open questions.