Representation Theory Of Algebras Research Articles

Boltje-Maisch resolutions of Specht modules

In \cite{21}, Boltje and Maisch found a permutation complex of Specht modules in representation theory of Hecke algebras, which is the same as the Boltje-Hartmann complex appeared in the representation theory of symmetric groups and general linear groups. In this paper we prove the exactness of Boltje-Maisch complex in the dominant weight case.

A Framework for Coxeter Spectral Analysis of Edge-bipartite Graphs, their Rational Morsifications and Mesh Geometries of Root Orbits

Following the spectral Coxeter analysis of matrix morsifications for Dynkin diagrams, the spectral graph theory, a graph coloring technique, and algebraic methods in graph theory, we continue our study of the category 𝒰Bigrn of loop-free edge-bipartite signed graphs Δ, with n ≥ 2 vertices, by means of the Coxeter number cΔ, the Coxeter spectrum speccΔ of Δ, that is, the spectrum of the Coxeter polynomial coxΔt ∈ $\mathbb{Z}$[t] and the $\mathbb{Z}$-bilinear Gram form bΔ : $\mathbb{Z}$n × $\mathbb{Z}$n → $\mathbb{Z}$ of Δ [SIAM J. Discrete Math. 272013]. Our main inspiration for the study comes from the representation theory of posets, groups and algebras, Lie theory, and Diophantine geometry problems. We show that the Coxeter spectral classification of connected edge-bipartite graphs Δ in 𝒰Bigrn reduces to the Coxeter spectral classification of rational matrix morsifications A ∈ $\widehat{M}$orDΔ for a simply-laced Dynkin diagram DΔ associated with Δ. Given Δ in 𝒰Bigrn, we study the isotropy subgroup Gln,$\mathbb{Z}$Δ of Gln, $\mathbb{Z}$ that contains the Weyl group $\mathbb{W}$Δ and acts on the set $\widehat{M}$orΔ of rational matrix morsifications A of Δ in such a way that the map A $\mapsto$ speccA, det A, cΔ is Gln, $\mathbb{Z}$Δ-invariant. It is shown that, for n ≤ 6, speccΔ is the spectrum of one of the Coxeter polynomials listed in Tables 3.11-3.11a we determine them by computer search using symbolic and numeric computation. The question, if two connected positive edge-bipartite graphs Δ,Δ' in 𝒰Bigrn, with speccΔ = speccΔ', are $\mathbb{Z}$-bilinear equivalent, is studied in the paper. The problem if any $\mathbb{Z}$-invertible matrix A ∈ Mn$\mathbb{Z}$ is $\mathbb{Z}$-congruent with its transpose Atr is also discussed.

Exotic nilpotent cones and representation theory of affine Hecke algebras of classical types

Exotic nilpotent cones and representation theory of affine Hecke algebras of classical types

Foundations of Discrete Optimization: In Transition from Linear to Non-linear Models and Methods

Optimization is a vibrant growing area of Applied Mathematics. Its many successful applications depend on efficient algorithms and this has pushed the development of theory and software. In recent years there has been a resurgence of interest to use “non-standard” techniques to estimate the complexity of computation and to guide algorithm design. New interactions with fields like algebraic geometry, representation theory, number theory, combinatorial topology, algebraic combinatorics, and convex analysis have contributed non-trivially to the foundations of computational optimization. In this expository survey we give three example areas of optimization where “algebraic-geometric thinking” has been successful. One key point is that these new tools are suitable for studying models that use non-linear constraints together with combinatorial conditions.

Higher-dimensional cluster combinatorics and representation theory

Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite-dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate a connection between these two seemingly unrelated subjects. We study triangulations of even-dimensional cyclic polytopes and tilting modules for higher Auslander algebras of linearly oriented type A which are summands of the cluster tilting module. We show that such tilting modules correspond bijectively to triangulations. Moreover mutations of tilting modules correspond to bistellar flips of triangulations. For any d -representation finite algebra we introduce a certain d -dimensional cluster category and study its cluster tilting objects. For higher Auslander algebras of linearly oriented type A we obtain a similar correspondence between cluster tilting objects and triangulations of a certain cyclic polytope. Finally we study certain functions on generalized laminations in cyclic polytopes, and show that they satisfy analogues of tropical cluster exchange relations. Moreover we observe that the terms of these exchange relations are closely related to the terms occurring in the mutation of cluster tilting objects.

Lie Algebras of Derivations and Resolvent Algebras

This paper analyzes the action {\delta} of a Lie algebra X by derivations on a C*-algebra A. This action satisfies an "almost inner" property which ensures affiliation of the generators of the derivations {\delta} with A, and is expressed in terms of corresponding pseudo-resolvents. In particular, for an abelian Lie algebra X acting on a primitive C*-algebra A, it is shown that there is a central extension of X which determines algebraic relations of the underlying pseudo- resolvents. If the Lie action {\delta} is ergodic, i.e. the only elements of A on which all the derivations in {\delta}_x vanish are multiples of the identity, then this extension is given by a (non-degenerate) symplectic form {\sigma} on X. Moreover, the algebra generated by the pseudo-resolvents coincides with the resolvent algebra based on the symplectic space (X, {\sigma}). Thus the resolvent algebra of the canonical commutation relations, which was recently introduced in physically motivated analyses of quantum systems, appears also naturally in the representation theory of Lie algebras of derivations acting on C*-algebras.

A probabilistic interpretation of the Macdonald polynomials

The two-parameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum polynomials. The Markov chain has stationary distribution a new two-parameter family of measures on partitions, the inverse of the Macdonald weight (rescaled). The uniform distribution on cycles of permutations and the Ewens sampling formula are special cases. The Markov chain is a version of the auxiliary variables algorithm of statistical physics. Properties of the Macdonald polynomials allow a sharp analysis of the running time. In natural cases, a bounded number of steps suffice for arbitrarily large k.

On the hyperdeterminant for 2×2×3 arrays

We use the representation theory of Lie algebras and computational linear algebra to determine the simplest non-constant invariant polynomial in the entries of a general 2 × 2 × 3 array. This polynomial is homogeneous of degree 6 and has 66 terms with coefficients ±1, ±2 in the 12 indeterminates x ijk where i, j = 1, 2 and k = 1, 2, 3. This invariant can be regarded as a natural generalization of Cayley's hyperdeterminant for 2 × 2 × 2 arrays.

Valued Graphs and the Representation Theory of Lie Algebras

Quivers (directed graphs), species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K -species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K -species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

In this paper we present Affine.m—a program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. The algorithms are based on the properties of weights and Weyl symmetry. Computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition are the most important problems for us. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in the popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras. Program summaryProgram title: Affine.mCatalogue identifier: AENA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, UKLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 24844No. of bytes in distributed program, including test data, etc.: 1045908Distribution format: tar.gzProgramming language: Mathematica.Computer: i386–i686, x86_64.Operating system: Linux, Windows, Mac OS, Solaris.RAM: 5–500 MbClassification: 4.2, 5.Nature of problem: Representation theory of finite-dimensional Lie algebras has many applications in different branches of physics, including elementary particle physics, molecular physics, nuclear physics. Representations of affine Lie algebras appear in string theories and two-dimensional conformal field theory used for the description of critical phenomena in two-dimensional systems. Also Lie symmetries play a major role in a study of quantum integrable systems.Solution method:We work with weights and roots of finite-dimensional and affine Lie algebras and use Weyl symmetry extensively. Central problems which are the computations of weight multiplicities, branching and fusion coefficients are solved using one general recurrent algorithm based on generalization of Weyl character formula. We also offer alternative implementation based on the Freudenthal multiplicity formula which can be faster in some cases.Restrictions:Computational complexity grows fast with the rank of an algebra, so computations for algebras of ranks greater than 8 are not practical.Unusual features:We offer the possibility of using a traditional mathematical notation for the objects in representation theory of Lie algebras in computations if Affine.m is used in the Mathematica notebook interface.Running time:From seconds to days depending on the rank of the algebra and the complexity of the representation.

Homotopy Theory

This workshop was a forum to present and discuss the latest result and ideas in homotopy theory and the connections to other branches of mathematics, such as algebraic geometry, representation theory and group theory.

Generalised Burnside rings, G-categories and module categories

This note describes an application of the theory of generalised Burnside rings to algebraic representation theory. Tables of marks are given explicitly for the groups S4 and S5 which are of particular interest in the context of reductive algebraic groups. As an application, the base sets for the nilpotent element F4(a3) are computed.

Representation theory of three-dimensional Sklyanin algebras

We determine the dimensions of the irreducible representations of the Sklyanin algebras with global dimension 3. This contributes to the study of marginal deformations of the N=4 super Yang–Mills theory in four dimensions in supersymmetric string theory. Namely, the classification of such representations is equivalent to determining the vacua of the aforementioned deformed theories.We also provide the polynomial identity degree for the Sklyanin algebras that are module finite over their center. The Calabi–Yau geometry of these algebras is also discussed.

Cluster-additive functions on stable translation quivers

Additive functions on translation quivers have played an important role in the representation theory of finite-dimensional algebras, the most prominent ones are the hammock functions introduced by S. Brenner. When dealing with cluster categories (and cluster-tilted algebras), one should look at a corresponding class of functions defined on stable translation quivers, namely the cluster-additive ones. We conjecture that the cluster-additive functions on a stable translation quiver of Dynkin type $\mathbb{A}_{n}, \mathbb{D}_{n}, \mathbb{E}_{6}, \mathbb {E}_{7}, \mathbb{E}_{8}$ are non-negative linear combinations of cluster-hammock functions (with index set a tilting set). The present paper provides a first study of cluster-additive functions and gives a proof of the conjecture in the case $\mathbb{A}_{n}$ .

Silting mutation in triangulated categories

In representation theory of algebras the notion of `mutation' often plays important roles, and two cases are well known, i.e. `cluster tilting mutation' and `exceptional mutation'. In this paper we focus on `tilting mutation', which has a disadvantage that it is often impossible, i.e. some of summands of a tilting object can not be replaced to get a new tilting object. The aim of this paper is to take away this disadvantage by introducing `silting mutation' for silting objects as a generalization of `tilting mutation'. We shall develope a basic theory of silting mutation. In particular, we introduce a partial order on the set of silting objects and establish the relationship with `silting mutation' by generalizing the theory of Riedtmann-Schofield and Happel-Unger. We show that iterated silting mutation act transitively on the set of silting objects for local, hereditary or canonical algebras. Finally we give a bijection between silting subcategories and certain t-structures.

Decomposition numbers for Hecke algebras of type G (r , p , n ): the (ε, q )-separated case

The paper studies the modular representation theory of the cyclotomic Hecke algebras of type G(r, p, n) with (ε, q)-separated parameters. We show that the decomposition numbers of these algebras are completely determined by the decomposition matrices of related cyclotomic Hecke algebras of type G(s, 1, m), where 1⩽s⩽r and 1⩽m⩽n. Furthermore, the proof gives an explicit algorithm for computing these decomposition numbers. Consequently, in principle, the decomposition matrices of these algebras are now known in characteristic zero. In proving these results, we develop a Specht module theory for these algebras, explicitly construct their simple modules and introduce and study analogues of the cyclotomic Schur algebras of type G(r, p, n) when the parameters are (ε, q)-separated. The main results of the paper rest upon two Morita equivalences: the first reduces the calculation of all decomposition numbers to the case of the l-splittable decomposition numbers and the second Morita equivalence allows us to compute these decomposition numbers using an analogue of the cyclotomic Schur algebras for the Hecke algebras of type G(r, p, n).

A generalized Cayley–Hamilton theorem

Let F be a solvable Lie subalgebra of the Lie algebra gln(C) (=Cn×n as a vector space). Let fk(x1,x2,…,xp),(k=1,2,…,r), be polynomials in the commuting variables x1,x2,…,xp with coefficients in C. For n×n matrices M1,M2,…,Mr, let F(x1,x2,…,xp)=∑k=1rMkfk(x1,x2,…,xp) and letδF(x1,x2,…,xp)=detF(x1,x2,…,xp).In this paper, we prove that, for A1,A2,…,Ap,M1,M2,…,Mr∈F, if one value of the matrix-valued function F(A1,A2,…,Ap) (the value depends on the product order of the variables) is nilpotent, then, (a) all values of F(A1,A2,…,Ap) are nilpotent; (b) all values of δF(A1,A2,…,Ap) (again depends on the product order of the variables) are nilpotent, and one value is 0. This generalizes the recent result in [7] and makes his result accurate. The main tool we use in this paper is the representation theory of solvable Lie algebras.

Some variants of Vaught's conjecture from the perspective of algebraic logic

Vaught’s Conjecture states that if T is a complete first order theory in a countable language such that T has uncountably many pairwise non-isomorphic countably infinite models, then T has 2^ℵ_0 many pairwise non-isomorphic countably infinite models. Continuing investigations initiated in S´agi, we apply methods of algebraic logic to study some variants of Vaught’s conjecture. More concretely, let S be a σ-compact monoid of selfmaps of the the natural numbers. We prove, among other things, that if a complete first order theory T has at least ℵ1 many countable models that cannot be elementarily embedded into each other by elements of S, then, in fact, T has continuum many such models. We also study-related questions in the context of equality free logics and obtain similar results. Our proofs are based on the representation theory of cylindric and quasi-polyadic algebras (for details see Henkin, Monk and Tarski (cylindric Algebras Part 1 and Part 2)) and topological properties of the Stone spaces of these algebras.

Handsaw Quiver Varieties and Finite W-Algebras

Following Braverman-Finkelberg-Feigin-Rybnikov (arXiv:1008.3655), we study the convolution algebra of a handsaw quiver variety, a.k.a. a parabolic Laumon space, and a finite W-algebra of type A. This is a finite analog of the AGT conjecture on 4-dimensional supersymmetric Yang-Mills theory with surface operators. Our new observation is that the C^*-fixed point set of a handsaw quiver variety is isomorphic to a graded quiver variety of type A, which was introduced by the author in connection with the representation theory of a quantum affine algebra. As an application, simple modules of the W-algebra are described in terms of IC sheaves of graded quiver varieties of type A, which were known to be related to Kazhdan-Lusztig polynomials. This gives a new proof of a conjecture by Brundan-Kleshchev on composition multiplicities on Verma modules, which was proved by Losev, in a wider context, by a different method.

Symplectic Reflection Algebras and Affine Lie Algebras

These are the notes of my talk at the conference “Double affine Hecke algebras and algebraic geometry” (MIT, May 18, 2010). The goal of this talk is to discuss some results and conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products categorifies certain structures in the representation theory for affine Lie algebras. These conjectures arose from the insight due to R. Bezrukavnikov and A. Okounkov on the link between quantum connections for Hilbert schemes of resolutions of Kleinian singularities and representations of symplectic reflection algebras, and took a much more definite shape after my conversations with I. Losev. At the moment, these notes are in a very preliminary form. Many proofs are just sketched or have missing details. The notes may (and probably do) contain errors, and should be viewed as no more than a basis for further thinking about these things. The plan is that these notes will be improved as I continue thinking about these questions, and as I receive feedback. So any feedback is very welcome! Acknowledgements. I would never have thought of these things without the encouragement and the vision of R. Bezrukavnikov and A. Okounkov. Also, I am very grateful to R. Bezrukavnikov and I. Losev for numerous discussions, without which I would have gotten nowhere. This work was supported by the NSF grant DMS-0854764.