arXiv · 0705.3903
We are going to determine all the self-injective cluster tilted algebras. All are of finite representation type and special biserial.
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Claus Michael Ringel. 2007-05-26. The Self-injective Cluster Tilted Algebras. https://arxiv.org/abs/0705.3903
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We prove that the polynomials counting locally free, absolutely indecomposable, rank 1 representations of quivers over rings of truncated power series have non-negative coefficients. This generalizes the positivity of toric Kac polynomials to higher depth. The proof goes by inductively contracting/deleting arrows of the quiver and is inspired by a previous work of Abdelgadir, Mellit and Rodriguez-Villegas on toric Kac polynomials. We also relate counts of absolutely indecomposable quiver representations in higher depth and counts of jets over fibres of quiver moment maps. This is expressed as a plethystic identity involving generating series of these counts. In rank 1, we prove a cohomological upgrade of this identity, by computing the compactly supported cohomology of jet spaces over preprojective stacks. This is reminiscent of PBW isomorphisms for preprojective cohomological Hall algebras. Finally, our plethystic identity allows us to prove two conjectures by Wyss on the asymptotic behavior of both counts, when depth goes to infinity.
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Let $K$ be a non-archimedean local field of residual characteristic $p\neq 2$. Let $G$ be a connected reductive group over $K$, let $θ$ be an involution of $G$ over $K$, and let $H$ be the connected component of $θ$-fixed subgroup of $G$ over $K$. By realizing the Steinberg representation of $G$ as the $G$-space of complex smooth harmonic cochains following the idea of Broussous--Courtès, we study its space of distinction by $H$ as a finite dimensional complex vector space. We give an upper bound of the dimension, and under certain conditions, we show that the upper bound is sharp by explicitly constructing a basis using the technique of Poincaré series. Finally, we apply our general theory to the case where $G$ is a general linear group and $H$ a special orthogonal subgroup, which leads to a complete classification result.