For each positive integer $n \geq 4$, we give an inequality satisfied by rank functions of arrangements of $n$ subspaces. When $n=4$ we recover Ingleton's inequality; for higher $n$ the inequalities are all new. These inequalities can be thought of as a hierarchy of necessary conditions for a (poly)matroid to be realizable. Some related open questions about the "cone of realizable polymatroids" are also presented.Comment: 10 pages, comments welcome. v2: correction to proof of Prop. 3, improved "Future directions" section, other minor improvements. v3: final version, minor change
We classify Hopf actions of Taft algebras T(n) on path algebras of quivers, in the setting where the quiver is loopless, finite, and Schurian. As a corollary, we see that every quiver admitting a faithful Z_n-action (by directed graph automorphisms) also admits inner faithful actions of a Taft algebra. Several examples for actions of the Sweedler algebra T(2) and for actions of T(3) are presented in detail. We then extend the results on Taft algebra actions on path algebras to actions of the Frobenius-Lusztig kernel u_q(sl_2), and to actions of the Drinfeld double of T(n).Comment: 29 pages. v3: Title changed, and Section 8 is ne
Abstract. In this paper we seek geometric and invariant-theoretic characterizations of (Schur-)representation finite algebras. To this end, we introduce two classes of finitedimensional algebras: those with the dense-orbit property and those with the multiplicityfree property. We show first that when a connected algebra A admits a preprojective component, each of these properties is equivalent to A being representation-finite. Next, we give an example of an algebra which is not representation-finite but still has the dense-orbit property. We also show that the string algebras with the dense orbit-property are precisely the representation-finite ones. Finally, we show that a tame algebra has the multiplicity-free property if and only if it is Schur-representation-finite.
Abstract. We describe a closed immersion from each representation space of a type A quiver with bipartite (i.e., alternating) orientation to a certain opposite Schubert cell of a partial flag variety. This "bipartite Zelevinsky map" restricts to an isomorphism from each orbit closure to a Schubert variety intersected with the above-mentioned opposite Schubert cell. For type A quivers of arbitrary orientation, we give the same result up to some factors of general linear groups.These identifications allow us to recover results of Bobiński and Zwara; namely we see that orbit closures of type A quivers are normal, Cohen-Macaulay, and have rational singularities. We also see that each representation space of a type A quiver admits a Frobenius splitting for which all of its orbit closures are compatibly Frobenius split.
Abstract. The free abelian group R(Q) on the set of indecomposable representations of a quiver Q, over a field K, has a ring structure where the multiplication is given by the tensor product. We show that if Q is a rooted tree (an oriented tree with a unique sink), then the ring R(Q) red is a finitely generated Z-module (here R(Q) red is the ring R(Q) modulo the ideal of all nilpotent elements). We will describe the ring R(Q) red explicitly, by studying functors from the category Rep(Q) of representations of Q over K to the category of finite dimensional K-vector spaces. IntroductionA quiver is just a directed graph Q = (Q • , Q → , t, h), where Q • is a vertex set, Q → is an arrow set, and t, h are functions from Q → to Q • giving the tail and head of an arrow, respectively. We assume Q • and Q → are finite in this paper. For any quiver Q and field K, there is a category Rep K (Q) of representations of Q over K. An object V of Rep K (Q) is an assignment of a finite dimensional K-vector space V x to each vertex x ∈ Q • , and an assignment of a K-linear map V a : V ta → V ha to each arrow a ∈ Q → . For any path p in Q, we get a K-linear map V p by composition. Morphisms in Rep K (Q) are given by linear maps at each vertex which form commutative diagrams over each arrow; see the book of Assem, Simson, and Skowroński [ASS06] for a precise definition of morphisms, and other fundamentals of quiver representations. We will fix some arbitrary field K throughout the paper and hence omit it from notation when possible.There is also a natural tensor product of quiver representations, induced by the tensor product in the category of vector spaces (cf. [Str00,Her08b]). Concretely, it is the "pointwise" tensor product of representations defined by (V ⊗ W ) x := V x ⊗ W x for each vertex x, and similarly for arrows. This tensor product gives the category Rep(Q) the structure of a tensor category in the sense of [DM82], and, along with direct sum, endows the set of isomorphism classes in Rep(Q) with a semiring structure. The associated ring R(Q) is the representation ring of Q (cf. §4), which is commutative with identity I Q , where we define (I Q ) x := K and (I Q ) a := id for all vertices x and arrows a. For a quiver Q which is not of Dynkin or extended Dynkin type, the problem of classifying its indecomposable representations is unsolved and very difficult, to say the least. Such a quiver is said to be of "wild representation type" (cf. [Dro80]) and has families of indecomposable representations depending on arbitrarily large numbers of parameters. This limits the effectiveness of an enumerative approach to studying tensor products of quiver representations (as opposed to, say, tensor products of representations of finite groups or the classical groups). Alternatively, we seek to describe R(Q) in abstract terms, and translate properties of R(Q) into properties of the tensor product in Rep(Q). For example, the main result of this paper has two equivalent formulations, the first of which (Theorem 39) can be stated in a simpl...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.