The global dimension of the full transformation monoid with an appendix by V. Mazorchuk and B. Steinberg

TL;DR

This paper determines the global dimension of the full transformation monoid algebra for all n and constructs explicit minimal projective resolutions, extending previous partial results and analyzing tilting modules.

Contribution

It proves the global dimension of the full transformation monoid algebra is n-1 for all n and provides explicit minimal projective resolutions.

Findings

Global dimension is n-1 for all n≥1.

Constructed explicit minimal projective resolutions of the trivial module.

Computed indecomposable tilting modules and Ringel duals.

Abstract

The representation theory of the symmetric group has been intensively studied for over 100 years and is one of the gems of modern mathematics. The full transformation monoid $\mathfrak T_n$ (the monoid of all self-maps of an $n$-element set) is the monoid analogue of the symmetric group. The investigation of its representation theory was begun by Hewitt and Zuckerman in 1957. Its character table was computed by Putcha in 1996 and its representation type was determined in a series of papers by Ponizovski{\u\i}, Putcha and Ringel between 1987 and 2000. From their work, one can deduce that the global dimension of $\mathbb C\mathfrak T_n$ is $n-1$ for $n=1,2,3,4$. We prove in this paper that the global dimension is $n-1$ for all $n\geq 1$ and, moreover, we provide an explicit minimal projective resolution of the trivial module of length $n-1$. In an appendix with V.~Mazorchuk we compute the indecomposable tilting modules of $\mathbb C\mathfrak T_n$ with respect to Putcha's quasi-hereditary structure and the Ringel dual (up to Morita equivalence).