Support $\tau$-tilting posets and Hochschild reconstruction for matrix centralizer algebras

Abstract

Let $R$ be a field, and let $\mathcal{O}$ be a finite-dimensional commutative local principal ideal $R$-algebra of Loewy length $\ell$ with $\mathrm{rad}\mathcal{O}=(\pi )$. For a nonempty set $P=\{p_1<\cdots <p_s\}\subseteq \{1,\dots ,\ell \}$, set ${\Lambda }_{\mathcal{O}}(P):={\mathrm{End}}_{\mathcal{O}}({⨁}_{p\in P}\mathcal{O}/({\pi }^p))$. We prove that the support $\tau$-tilting poset of ${\Lambda }_{\mathcal{O}}(P)$ is isomorphic to the left weak order on ${\Sigma }_{s+1}$ and hence recovers precisely $s$. We also show that $Z({\Lambda }_{\mathcal{O}}(P))\simeq \mathcal{O}/({\pi }^{p_s})$ and, as a module over this center, $H{H}_0({\Lambda }_{\mathcal{O}}(P))\simeq {⨁}_{i=1}^s\mathcal{O}/({\pi }^{g_i})$, where $g_1=p_1$ and $g_i=p_i-p_{i-1}$ for $i\ge 2$. Applied blockwise, these formulas compute the corresponding invariants of matrix centralizer algebras over arbitrary fields. Finally, a polynomial primary block is Morita equivalent to a split string algebra if and only if its defining irreducible polynomial is linear and its exponent set is $\{p\}$ or $\{p,p+1\}$. It is Morita equivalent to a split gentle algebra if and only if the polynomial is linear and the exponent set is $\{1\}$, $\{2\}$, or $\{1,2\}$. These criteria yield Morita reconstruction within the corresponding classes.

32 pages