We give a description of faces, of all codimensions, for the cones spanned by the set of weights associated to the rings of semi-invariants of quivers. For a triple flag quiver and its faces of codimension 1 this description reduces to the result of Knutson-Tao-Woodward on the facets of the Klyachko cone. We give new applications to Littlewood-Richardson coefficients, including a product formula for LR-coefficients corresponding to triples of partitions lying on a wall of the Klyachko cone. We systematically...

We prove that a completely separating incidence algebra of a partially ordered set is of tame representation type if and only if the associated Tits integral quadratic form is weakly non-negative.

We prove that the component quiver ${\Sigma }_A$ of a connected self-injective artin algebra A of infinite representation type is fully cyclic, that is, every finite set of components of the Auslander-Reiten quiver ${\Gamma }_A$ of A lies on a common oriented cycle in ${\Sigma }_A$.

We study when the composite of n irreducible morphisms between modules in a regular component of the Auslander-Reiten quiver is non-zero and lies in the n+1-th power of the radical ℜ of the module category. We prove that in this case such a composite belongs to ${\Re }^{\infty }$. We apply these results to characterize those string algebras having n irreducible morphisms between band modules such that their composite is a non-zero morphism in ${\Re }^{n+1}$.

Let $Q$ be a finite union of Dynkin quivers, $G\subseteq \mathrm{Aut}\left(𝕜Q\right)$ a finite abelian group, $\widehat{Q}$ the generalized McKay quiver of $(Q,G)$ and ${\Gamma }_Q$ the Auslander-Reiten quiver of $𝕜Q$. Then $G$ acts functorially on the quiver ${\Gamma }_Q$. We show that the Auslander-Reiten quiver of $𝕜\widehat{Q}$ coincides with the generalized McKay quiver of $({\Gamma }_Q,G)$.

Let $𝕜$ be an algebraically closed field of characteristic $p\ne 2$, and let $Q_8$ be the quaternion group. We describe the structures of all simple modules over the quantum double $D\left(𝕜Q_8\right)$ of group algebra $𝕜Q_8$. Moreover, we investigate the tensor product decomposition rules of all simple $D\left(𝕜Q_8\right)$-modules. Finally, we describe the Grothendieck ring $G_0\left(D\left(𝕜Q_8\right)\right)$ by generators with relations.

This paper studies the Hochschild cohomology of finite-dimensional monomial algebras. If Λ = K/I with I an admissible monomial ideal, then we give sufficient conditions for the existence of an embedding of $K[x₁,...,x_r]/⟨x_a{x}_{b}fora\ne b⟩$ into the Hochschild cohomology ring HH*(Λ). We also introduce stacked algebras, a new class of monomial algebras which includes Koszul and D-Koszul monomial algebras. If Λ is a stacked algebra, we prove that $HH*\left(\Lambda \right)/\cong K[x₁,...,x_r]/⟨x_a{x}_{b}fora\ne b⟩$, where is the ideal in HH*(Λ) generated by the homogeneous nilpotent elements. In...

Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector $m\left(M\right)={\left(m_x\right)}_{x\in X}\in {\mathbb{N}}^X$ such that $M\cong {⨁}_{x\in X}{X}_x^{m_x}$ is studied. A precise formula for $di{m}_{k}Ho{m}_{\Lambda }(M,X)$, for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors in Colloq....

Given a module M over an algebra Λ and a complete set of pairwise nonisomorphic indecomposable Λ-modules, the problem of determining the vector $m\left(M\right)={\left(m_X\right)}_{X\in }\in {\mathbb{N}}^{}$ such that $M\cong {⨁}_{X\in }{X}^{m_X}$ is studied. A general method of finding the vectors m(M) is presented (Corollary 2.1, Theorem 2.2 and Corollary 2.3). It is discussed and applied in practice for two classes of algebras: string algebras of finite representation type and hereditary algebras of type ${̃}_{p,q}$. In the second case detailed algorithms are given (Algorithms 4.5 and 5.5).