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 Algebras and Representation Theory   [SJR: 0.868]   [H-I: 20]   [1 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1572-9079 - ISSN (Online) 1386-923X    Published by Springer-Verlag  [2355 journals]
• Degenerations of Submodules and Composition Series
• Authors: Nils M. Nornes; Steffen Oppermann
Pages: 803 - 819
Abstract: Let M and N be modules over an artin algebra such that M degenerates to N. We show that any submodule of M degenerates to a submodule of N. This suggests that a composition series of M will in some sense degenerate to a composition series of N. We then study a subvariety of the module variety, consisting of those representations where all matrices are upper triangular. We show that these representations can be seen as representations of composition series, and that the orbit closures describe the above mentioned degeneration of composition series.
PubDate: 2017-08-01
DOI: 10.1007/s10468-017-9677-0
Issue No: Vol. 20, No. 4 (2017)

• Branching Rules for Finite-Dimensional U q ( ð–˜ ð–š ( 3 ) )
$\mathcal {U}_{q}(\mathfrak {su}(3))$ -Representations with Respect to a
Right Coideal Subalgebra
• Authors: Noud Aldenhoven; Erik Koelink; Pablo Román
Pages: 821 - 842
Abstract: We consider the quantum symmetric pair $$(\mathcal {U}_{q}(\mathfrak {su}(3)), \mathcal {B})$$ where $$\mathcal {B}$$ is a right coideal subalgebra. We prove that all finite-dimensional irreducible representations of $$\mathcal {B}$$ are weight representations and are characterised by their highest weight and dimension. We show that the restriction of a finite-dimensional irreducible representation of $$\mathcal {U}_{q}(\mathfrak {su}(3))$$ to $$\mathcal {B}$$ decomposes multiplicity free into irreducible representations of $$\mathcal {B}$$ . Furthermore we give explicit expressions for the highest weight vectors in this decomposition in terms of dual q-Krawtchouk polynomials.
PubDate: 2017-08-01
DOI: 10.1007/s10468-017-9678-z
Issue No: Vol. 20, No. 4 (2017)

• Whittaker Modules for the Insertion-Elimination Lie Algebra
• Authors: Matthew Ondrus; Emilie Wiesner
Pages: 843 - 856
Abstract: This paper addresses the representation theory of the insertion-elimination Lie algebra, a Lie algebra that can be naturally realized in terms of tree-inserting and tree-eliminating operations on rooted trees. The insertion-elimination algebra admits a triangular decomposition in the sense of Moody and Pianzola, and thus it is natural to define Whittaker modules corresponding to a given algebra homomorphism. Among other results, we show that the standard Whittaker modules are simple under certain constraints on the corresponding algebra homomorphism.
PubDate: 2017-08-01
DOI: 10.1007/s10468-016-9665-9
Issue No: Vol. 20, No. 4 (2017)

• Derived Equivalence Classification of the Gentle Two-Cycle Algebras
• Authors: Grzegorz Bobinski
Pages: 857 - 869
Abstract: We complete a derived equivalence classification of the gentle two-cycle algebras initiated in earlier papers by Avella-Alaminos and Bobinski–Malicki.
PubDate: 2017-08-01
DOI: 10.1007/s10468-016-9666-8
Issue No: Vol. 20, No. 4 (2017)

• Description of B ( ∞ ) $\mathsf {B}(\infty )$ through Kashiwara
Embedding for Lie Algebra Types E 6 and E 7
• Authors: Jin Hong; Hyeonmi Lee
Pages: 871 - 893
Abstract: We study the crystal base $$\mathsf {B}(\infty )$$ associated with the negative part of the quantum group for finite simple Lie algebras of types E 6 and E 7. We present an explicit description of $$\mathsf {B}(\infty )$$ as the image of a Kashiwara embedding that is in natural correspondence with the marginally large tableau description of $$\mathsf {B}(\infty )$$ .
PubDate: 2017-08-01
DOI: 10.1007/s10468-017-9667-2
Issue No: Vol. 20, No. 4 (2017)

• From Grassmann Necklaces to Restricted Permutations and Back Again
• Authors: Karel Casteels; Siân Fryer
Pages: 895 - 921
Abstract: We study the commutative algebras Z J K appearing in Brown and Goodearl’s extension of the $$\mathcal {H}$$ -stratification framework, and show that if A is the single parameter quantized coordinate ring of M m,n , G L n or S L n , then the algebras Z J K can always be constructed in terms of centres of localizations. The main purpose of the Z J K is to study the structure of the topological space s p e c(A), which remains unknown for all but a few low-dimensional examples. We explicitly construct the required denominator sets using two different techniques (restricted permutations and Grassmann necklaces) and show that we obtain the same sets in both cases. As a corollary, we obtain a simple formula for the Grassmann necklace associated to a cell of totally nonnegative real m × n matrices in terms of its restricted permutation.
PubDate: 2017-08-01
DOI: 10.1007/s10468-017-9668-1
Issue No: Vol. 20, No. 4 (2017)

• Geometric Realizations of Lusztig’s Symmetries of Symmetrizable
Quantum Groups
• Authors: Minghui Zhao
Pages: 923 - 950
Abstract: Let U be the quantum group and f be the Lusztig’s algebra associated with a symmetrizable generalized Cartan matrix. The algebra f can be viewed as the positive part of U. Lusztig introduced some symmetries T i on U for all i ∈ I. Since T i (f) is not contained in f, Lusztig considered two subalgebras i f and i f of f for any i ∈ I, where i f={x ∈ f T i (x) ∈ f} and $${^{i}\mathbf {f}}=\{x\in \mathbf {f}\,\, \,\,T^{-1}_{i}(x)\in \mathbf {f}\}$$ . The restriction of T i on i f is also denoted by $$T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}$$ . The geometric realization of f and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig’s symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of i f, i f and $$T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}$$ by using the language ’quiver with automorphism’ introduced by Lusztig.
PubDate: 2017-08-01
DOI: 10.1007/s10468-017-9669-0
Issue No: Vol. 20, No. 4 (2017)

• Decomposition of Tensor Products Involving a Steinberg Module
• Authors: Tobias Kildetoft
Pages: 951 - 975
Abstract: We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group G and when restricted to either a Frobenius kernel G r or a finite Chevalley group $$G(\mathbb {F}_q)$$ . In all three cases, we give formulas reducing this to standard character data for G. Along the way, we use a bilinear form on the characters of finite dimensional G-modules to give formulas for the dimension of homomorphism spaces between certain G-modules when restricted to either G r or $$G(\mathbb {F}_q)$$ . Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for G which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin’s tilting conjecture.
PubDate: 2017-08-01
DOI: 10.1007/s10468-017-9670-7
Issue No: Vol. 20, No. 4 (2017)

• Locally Quadratic Modules and Minuscule Representations
Pages: 977 - 1005
Abstract: We give a new, geometric proof of a theorem by Timmesfeld showing that for simple Chevalley groups, abstract modules where all roots act quadratically are direct sums of minuscule representations. Our proof is uniform, treats finite and infinite fields on an equal footing, and includes Lie rings.
PubDate: 2017-08-01
DOI: 10.1007/s10468-017-9671-6
Issue No: Vol. 20, No. 4 (2017)

• On Frobenius (Completed) Orbit Categories
• Authors: Alfredo Nájera Chávez
Pages: 1007 - 1027
Abstract: Let ð“” be a Frobenius category, $${\mathcal P}$$ its subcategory of projective objects and F : ð“” → ð“” an exact automorphism. We prove that there is a fully faithful functor from the orbit category ð“”/F into $$\operatorname {gpr}({\mathcal P}/F)$$ , the category of finitely-generated Gorenstein-projective modules over $${\mathcal P}/F$$ . We give sufficient conditions to ensure that the essential image of ð“”/F is an extension-closed subcategory of $$\operatorname {gpr}({\mathcal P}/F)$$ . If ð“” is in addition Krull-Schmidt, we give sufficient conditions to ensure that the completed orbit category $${\mathcal E} \ \widehat {\!\! /} F$$ is a Krull-Schmidt Frobenius category. Finally, we apply our results on completed orbit categories to the context of Nakajima categories associated to Dynkin quivers and sketch applications to cluster algebras.
PubDate: 2017-08-01
DOI: 10.1007/s10468-017-9672-5
Issue No: Vol. 20, No. 4 (2017)

• The Representation Type of Determinantal Varieties
• Authors: Jan O. Kleppe; Rosa M. Miró-Roig
Pages: 1029 - 1059
Abstract: This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves $$\mathcal {E}$$ of arbitrary high rank on a general standard (resp. linear) determinantal scheme $$X\subset \mathbb {P}^{n}$$ of codimension c ≥ 1, n − c ≥ 1 and defined by the maximal minors of a t × (t + c−1) homogeneous matrix $$\mathcal {A}$$ . The sheaves $$\mathcal {E}$$ are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme $$X\subset \mathbb {P}^{n}$$ is of wild representation type provided the degrees of the entries of the matrix $$\mathcal {A}$$ satisfy some weak numerical assumptions; and (2) we determine values of t, n and n − c for which a linear standard determinantal scheme $$X\subset \mathbb {P}^{n}$$ is of wild representation type with respect to the much more restrictive category of its indecomposable Ulrich sheaves, i.e. X is of Ulrich wild representation type.
PubDate: 2017-08-01
DOI: 10.1007/s10468-017-9673-4
Issue No: Vol. 20, No. 4 (2017)

• Off-Shell Supersymmetry and Filtered Clifford Supermodules
• Authors: Charles F. Doran; Michael G. Faux; Sylvester J. Gates; Tristan Hübsch; Kevin Iga; Gregory D. Landweber
Abstract: An off-shell representation of supersymmetry is a representation of the super Poincaré algebra on a dynamically unconstrained space of fields. We describe such representations formally, in terms of the fields and their spacetime derivatives, and we interpret the physical concept of engineering dimension as an integral grading. We prove that formal graded off-shell representations of one-dimensional N-extended supersymmetry, i.e., the super Poincaré algebra $$\mathfrak {p}^{1 N}$$ , correspond to filtered Clifford supermodules over Cl(N). We also prove that formal graded off-shell representations of two-dimensional (p,q)-supersymmetry, i.e., the super Poincaré algebra $$\mathfrak {p}^{1,1 p,q}$$ , correspond to bifiltered Clifford supermodules over Cl(p + q). Our primary tools are Rees superalgebras and Rees supermodules, the formal deformations of filtered superalgebras and supermodules, which give a one-to-one correspondence between filtered spaces and graded spaces with even degree-shifting injections. This generalizes the machinery used by Gerstenhaber to prove that every filtered algebra is a deformation of its associated graded algebra. Our treatment extends the notion of Rees algebras and modules to filtrations which are compatible with a supersymmetric structure. We also describe the analogous constructions for bifiltrations and bigradings.
PubDate: 2017-07-18
DOI: 10.1007/s10468-017-9718-8

• Structure of Irreducible Homomorphisms to/from Free Modules
• Authors: Saeed Nasseh; Ryo Takahashi
Abstract: The primary goal of this paper is to investigate the structure of irreducible monomorphisms to and irreducible epimorphisms from finitely generated free modules over a noetherian local ring. Then we show that over such a ring, self-vanishing of Ext and Tor for a finitely generated module admitting such an irreducible homomorphism forces the ring to be regular.
PubDate: 2017-07-13
DOI: 10.1007/s10468-017-9722-z

• Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups
• Authors: T. Geetha; Amritanshu Prasad
Abstract: Young’s orthogonal basis is a classical basis for an irreducible representation of a symmetric group. This basis happens to be a Gelfand-Tsetlin basis for the chain of symmetric groups. It is well-known that the chain of alternating groups, just like the chain of symmetric groups, has multiplicity-free restrictions for irreducible representations. Therefore each irreducible representation of an alternating group also admits Gelfand-Tsetlin bases. Moreover, each such representation is either the restriction of, or a subrepresentation of, the restriction of an irreducible representation of a symmetric group. In this article, we describe a recursive algorithm to write down the expansion of each Gelfand-Tsetlin basis vector for an irreducible representation of an alternating group in terms of Young’s orthogonal basis of the ambient representation of the symmetric group. This algorithm is implemented with the Sage Mathematical Software.
PubDate: 2017-07-12
DOI: 10.1007/s10468-017-9706-z

• Involutions of Type G 2 Over Fields of Characteristic Two
• Authors: John Hutchens; Nathaniel Schwartz
Abstract: We continue a study of automorphisms of order 2 of algebraic groups. In particular we look at groups of type G2 over fields k of characteristic two. Let C be an octonion algebra over k; then Aut(C) is a group of type G2 over k. We characterize automorphisms of order 2 and their corresponding fixed point groups for Aut(C) by establishing a connection between the structure of certain four dimensional subalgebras of C and the elements in Aut(C) that induce inner automorphisms of order 2. These automorphisms relate to certain quadratic forms which, in turn, determine the Galois cohomology of the fixed point groups of the involutions. The characteristic two case is unique because of the existence of four dimensional totally singular subalgebras. Over finite fields we show how our results coincide with known results, and we establish a classification of automorphisms of order 2 over infinite fields of characteristic two.
PubDate: 2017-07-12
DOI: 10.1007/s10468-017-9723-y

• Injective Presentations of Induced Modules over Cluster-Tilted Algebras
• Authors: Ralf Schiffler; Khrystyna Serhiyenko
Abstract: Every cluster-tilted algebra B is the relation extension $$C\ltimes \textup {Ext}^{2}_{C}(DC,C)$$ of a tilted algebra C. A B-module is called induced if it is of the form M⊗ C B for some C-module M. We study the relation between the injective presentations of a C-module and the injective presentations of the induced B-module. Our main result is an explicit construction of the modules and morphisms in an injective presentation of any induced B-module. In the case where the C-module, and hence the B-module, is projective, our construction yields an injective resolution. In particular, it gives a module theoretic proof of the well-known 1-Gorenstein property of cluster-tilted algebras.
PubDate: 2017-07-10
DOI: 10.1007/s10468-017-9721-0

• Limits of Multiplicities in Excellent Filtrations and Tensor Product
Decompositions for Affine Kac-Moody Algebras
• Authors: Dijana Jakelić; Adriano Moura
Abstract: We express the multiplicities of the irreducible summands of certain tensor products of irreducible integrable modules for an affine Kac-Moody algebra over a simply laced Lie algebra as sums of multiplicities in appropriate excellent filtrations (Demazure flags). As an application, we obtain expressions for the outer multiplicities of tensor products of two fundamental modules for $$\widehat {\mathfrak {sl}}_{2}$$ in terms of partitions with bounded parts, which subsequently lead to certain partition identities.
PubDate: 2017-07-08
DOI: 10.1007/s10468-017-9712-1

• Criteria for a Ring to have aLeft Noetherian Largest Left Quotient Ring
• Authors: V. V. Bavula
Abstract: Criteria are given for a ring to have a left Noetherian largest left quotient ring. It is proved that each such a ring has only finitely many maximal left denominator sets. An explicit description of them is given. In particular, every left Noetherian ring has only finitely many maximal left denominator sets.
PubDate: 2017-07-05
DOI: 10.1007/s10468-017-9717-9

• Parahoric Restriction for GSp(4)
• Authors: Mirko Rösner
Abstract: Parahoric restriction is the parahoric analogue of Jacquet’s functor. The group GSp(4, F) of symplectic similitudes of genus two over a local number field F/ℚ p has five conjugacy classes of parahoric subgroups. For each we determine the parahoric restriction of the non-cuspidal irreducible smooth representations of GSp(4, F) in terms of explicit character values.
PubDate: 2017-07-05
DOI: 10.1007/s10468-017-9707-y

• On Standard Derived Equivalences of Orbit Categories
• Authors: Yury Volkov; Alexandra Zvonareva
Abstract: Let k be a commutative ring, $$\mathcal {A}$$ and $$\mathcal {B}$$ – two k-linear categories with an action of a group G. We introduce the notion of a standard G-equivalence from $$\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$$ to $$\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$$ , where $$\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$$ is the homotopy category of finitely generated projective $$\mathcal {A}$$ -complexes. We construct a map from the set of standard G-equivalences to the set of standard equivalences from $$\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$$ to $$\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$$ and a map from the set of standard G-equivalences from $$\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$$ to $$\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$$ to the set of standard equivalences from $$\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {B}/G)$$ to $$\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {A}/G)$$ , where $$\mathcal {A}/G$$ denotes the orbit category. We investigate the properties of these maps and apply our results to the case where $$\mathcal {A}=\mathcal {B}=R$$ is a Frobenius k-algebra and G is the cyclic group generated by its Nakayama automorphism ν. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field.
PubDate: 2017-06-30
DOI: 10.1007/s10468-017-9710-3

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