Stable representations of quivers

Features for speech emotion recognition are usually dominated by the spectral magnitude information while they ignore the use of the phase spectrum because of the difficulty of properly interpreting it. Motivated by recent successes of phase-based features for speech processing, this paper investigates the effectiveness of phase information for whispered speech emotion recognition. We select two types of phase-based features (i.e., modified group delay features and all-pole group delay features), both which have shown wide applicability to all sorts of different speech analysis and are now studied in whispered speech emotion recognition. When exploiting these features, we propose a new speech emotion recognition framework, employing outer product in combination with power and L2 normalization. The according technique encodes any variable length sequence of the phase-based features into a fixed dimension vector regardless of the length of the input sequence. The resulting representation is fed to train a classification model with a linear kernel classifier. Experimental results on the Geneva Whispered Emotion Corpus database, including normal and whispered phonation, demonstrate the effectiveness of the proposed method when compared with other modern systems. It is also shown that, combining phase information with magnitude information could significantly improve performance over the common systems solely adopting magnitude information.

On dimension of poset variety

<p>We can view quiver representations of a fixed dimension vector as an algebraic variety over an algebraically closed field $K$. There is an action of the product of general linear groups on each of these varieties where the orbits of the action correspond to isomorphism classes of quiver representation. A $K$-algebra $A$ is said to have the dense orbit property if for each dimension vector, the product of the general linear group acts on each irreducible component of the module variety with a dense orbit. Under certain conditions, a $K$ algebra $A$ is representation finite if and only if it $A$ has the dense orbit property. The implication representation finite implies the dense orbit property is always true. The converse is not true in general, as shown by Chindris, Kinser, and Weyman in \cite{ryan}. Our main theorem of this thesis builds on their work to give a family of representation infinite algebras with the dense orbit property. We also give a conjectured classification of indecomposables with dense orbits. \par</p> <p>In the future, we hope the work presented here can be used to find even more examples of representation infinite algebra with the dense orbit property to then develop deeper theory to classify algebras with the dense orbit property that are representation infinite.</p>

Accumulation points of real Schur roots

Let G be a connected semi-simple Lie group with finite center, say — then it is not necessarily the case that G possesses enough finite dimensional representations to separate points (cf. 3.1.1). In fact G will, in general, admit no non-trivial finite dimensional unitary representations (cf. number 4.3.2). Nevertheless, despite these apparently discouraging facts, the finite dimensional representations of G are important — this will be elaborated on below.

We consider stable representations of non-Dynkin quivers with respect to a central charge. These attract a lot of interest in mathematics and physics since they can be identified with so-called BPS states. Another motivation is the work of Dimitrov et al. on the phases of stable representations of the generalized Kronecker quiver. One aim is to explain for general Euclidean and wild quivers the behavior of phases of stable representations well known in some examples. In addition, we study especially the behavior of preinjective, postprojective and regular indecomposable modules. We show that the existence of a stable representation with self-extensions implies the existence of infinitely many stables without self-extensions for rigid central charges. In this case the phases of the stable representations approach one or two limit points. In particular, the phases are not dense in two arcs. The category of representations of acyclic quivers is a special case of rigid Abelian categories which show this behavior for rigid central charges.

We compute the characters of the finite dimensional irreducible representations of the Lie superalgebra g l ( m | n ) \mathfrak {gl}(m|n) , and determine Ext {\operatorname {Ext}} ’s between simple modules in the category of finite dimensional representations. We formulate conjectures for the analogous results in category O \mathcal O . The combinatorics parallels the combinatorics of certain canonical bases over the Lie algebra g l ( ∞ ) \mathfrak {gl}(\infty ) .

The concept of modulation is generalized to pseudo-modulation and its subclasses including pre-modulation, generalized modulation and regular modulation. The motivation is to define the valued analogue of natural quiver, called {\em natural valued quiver}, of an artinian algebra so as to correspond to its valued Ext-quiver when this algebra is not $k$-splitting over the field $k$. Moreover, we illustrate the relation between the valued Ext-quiver and the natural valued quiver. The interesting fact we find is that the representation categories of a pseudo-modulation and of a pre-modulation are equivalent respectively to that of a tensor algebra of $\mathcal A$-path type and of a generalized path algebra. Their examples are given respectively from two kinds of artinian hereditary algebras. Furthermore, the isomorphism theorem is given for normal generalized path algebras with finite (acyclic) quivers and normal pre-modulations. Four examples of pseudo-modulations are given: (i) group species in mutation theory as a semi-normal generalized modulation; (ii) viewing a path algebra with loops as a pre-modulation with valued quiver which has not loops; (iii) differential pseudo-modulation and its relation with differential tensor algebras; (iv) a pseudo-modulation is considered as a free graded category.

We study degenerations of the Belavin R-matrices via the infinite dimensional operators defined by Shibukawa-Ueno. We define a two-parameter family of generalizations of the Shibukawa-Ueno R-operators. These operators have finite dimensional representations which include Belavin's R-matrices in the elliptic case, a two-parameter family of twisted affinized Cremmer-Gervais R-matrices in the trigonometric case, and a two-parameter family of twisted (affinized) generalized Jordanian R-matrices in the rational case. We find finite dimensional representations which are compatible with the elliptic to trigonometric and rational degeneration. We further show that certain members of the elliptic family of operators have no finite dimensional representations. These R-operators unify and generalize earlier constructions of Felder and Pasquier, Ding and Hodges, and the authors, and illuminate the extent to which the Cremmer-Gervais R-matrices (and their rational forms) are degenerations of Belavin's R-matrix.

This paper focuses on the finite dimensional irreducible representations of Lie superalgebra D(2, 1; α). We explicitly construct the finite dimensional representations of the superalgebra D(2, 1; α) by using the shift operator and differential operator representations. Unlike ordinary Lie algebra representation, there are typical and atypical representations for most superalgebras. Therefore, its typical and atypical representation conditions are also given. Our results are expected to be useful for the construction of primary fields of the corresponding current superalgebra of D(2, 1; α).

ABSTRACTIn this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and 𝒬 a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra R𝒬, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Quantized multiplicative quiver varieties

Supersymmetric D-branes supported on the complex two-dimensional base S of the local Calabi–Yau threefold \(K_S\) are described by semi-stable coherent sheaves on S. Under suitable conditions, the BPS indices counting these objects (known as generalized Donaldson–Thomas invariants) coincide with the Vafa–Witten invariants of S (which encode the Betti numbers of the moduli space of semi-stable sheaves). For surfaces which admit a strong collection of exceptional sheaves, we develop a general method for computing these invariants by exploiting the isomorphism between the derived category of coherent sheaves and the derived category of representations of a suitable quiver with potential (Q, W) constructed from the exceptional collection. We spell out the dictionary between the Chern class \(\gamma \) and polarization J on S versus the dimension vector \(\vec {N}\) and stability parameters \(\vec {\zeta }\) on the quiver side. For all examples that we consider, which include all del Pezzo and Hirzebruch surfaces, we find that the BPS indices \(\Omega _\star (\gamma )\) at the attractor point (or self-stability condition) vanish, except for dimension vectors corresponding to simple representations and pure D0-branes. This opens up the possibility to compute the BPS indices in any chamber using either the flow tree or the Coulomb branch formula. In all cases we find precise agreement with independent computations of Vafa–Witten invariants based on wall-crossing and blow-up formulae. This agreement suggests that (1) generating functions of DT invariants for a large class of quivers coming from strong exceptional collections are mock modular functions of higher depth and (2) non-trivial single-centered black holes and scaling solutions do not exist quantum mechanically in such local Calabi–Yau geometries.

On derived Hall numbers for tame quivers