The global dimension of the full transformation monoid (with an appendix by V. Mazorchuk and B. Steinberg) (Q295908)

Here, the representation theory of the full symmetric semigroup \(\mathfrak T_n\) of transformations of an \(n\)-element set is considered. It was known that the global dimension of \(\mathfrak{CT}_n\) for \(n\leqslant 4\) is \(n-1\). Here, the author proves that the global dimension is \(n - 1\) for all \(n \geqslant 1\), presents an explicit minimal projective resolution of the trivial module of length \(n - 1\) and computes the indecomposable tilting modules (modules that have both a standard filtration and a costandard filtration) of \(\mathfrak{CT}_n\) with respect to Putcha's quasi-hereditary structure and the Ringel dual (up to Morita equivalence).