# Degenerate 0-Schur algebras and Nil-Temperley-Lieb algebras

## TL;DR

This paper introduces degenerate 0-Schur algebras as quotients of 0-Schur algebras, explores their geometric interpretation via double flag varieties, and connects them to nil-Hecke and nil-Temperley-Lieb algebras.

## Contribution

It defines and studies degenerate 0-Schur algebras, revealing their structure as associated graded algebras and their relations to well-known algebraic structures.

## Key findings

- Degenerate 0-Schur algebras are quotients of 0-Schur algebras.
- They can be realized as associated graded algebras.
- Connections to nil-Hecke and nil-Temperley-Lieb algebras are established.

## Abstract

In \cite{JS} Jensen and Su constructed 0-Schur algebras on double flag varieties. The construction leads to a presentation of 0-Schur algebras using quivers with relations and the quiver approach naturally gives rise to a new class of algebras. That is, the path algebras defined on the quivers of 0-Schur algebras with relations modified from the defining relations of 0-Schur algebras by a tuple of parameters $\ut$. In particular, when all the entries of $\ut$ are 1, we have 0-Schur algerbas. When all the entries of $\ut$ are zero, we obtain a class of degenerate 0-Schur algebras. We prove that the degenerate algebras are associated graded algebras and quotients of 0-Schur algebras. Moreover, we give a geometric interpretation of the degenerate algebras using double flag varieties, in the same spirit as \cite{JS}, and show how the centralizer algebras are related to nil-Hecke algebras and nil-Temperly-Lieb algebras