Quasi-hereditary covers of Temperley-Lieb algebras and relative dominant dimension

DOI10.1017/PRM.2024.35arXiv2212.00099MaRDI QIDQ6508196

Tiago Cruz, Karin Erdmann

Abstract: Many connections and dualities in representation theory can be explained using quasi-hereditary covers in the sense of Rouquier. The concepts of relative dominant and codominant dimension with respect to a module, introduced recently by the first-named author, are important tools to evaluate and classify quasi-hereditary covers. In this paper, we prove that the relative dominant dimension of the regular module of a quasi-hereditary algebra with a simple preserving duality with respect to a summand $Q$ of a characteristic tilting module equals twice the relative dominant dimension of a characteristic tilting module with respect to $Q$. To resolve the Temperley-Lieb algebras of infinite global dimension, we apply this result to the class of Schur algebras $S(2,d)$ and $Q=V^{otimesd}$ the $d$-tensor power of the 2-dimensional module and we completely determine the relative dominant dimension of the Schur algebra $S(2,d)$ with respect to $V^{otimesd}$. The $q$-analogues of these results are also obtained. As a byproduct, we obtain a Hemmer-Nakano type result connecting the Ringel duals of $q$-Schur algebras and Temperley-Lieb algebras. From the point of view of Temperley-Lieb algebras, we obtain the first complete classification of their connection to their quasi-hereditary covers formed by Ringel duals of $q$-Schur algebras. These results are compatible with the integral setup, and we use them to deduce that the Ringel dual of a $q$-Schur algebra over the ring of Laurent polynomials over the integers together with some projective module is the best quasi-hereditary cover of the integral Temperley-Lieb algebra.

Mathematics Subject Classification ID

Representations of orders, lattices, algebras over commutative rings (16G30) Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics (82B20) Representations of associative Artinian rings (16G10) Homological dimension in associative algebras (16E10) Schur and (q)-Schur algebras (20G43)

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