Authors:Nils M. Nornes; Steffen Oppermann Pages: 803 - 819 Abstract: 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)

Authors:Noud Aldenhoven; Erik Koelink; Pablo Román Pages: 821 - 842 Abstract: 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)

Authors:Matthew Ondrus; Emilie Wiesner Pages: 843 - 856 Abstract: 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)

Authors:Jin Hong; Hyeonmi Lee Pages: 871 - 893 Abstract: 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)

Authors:Karel Casteels; Siân Fryer Pages: 895 - 921 Abstract: 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)

Authors:Minghui Zhao Pages: 923 - 950 Abstract: 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)

Authors:Tobias Kildetoft Pages: 951 - 975 Abstract: 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)

Authors:Adrien Deloro 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)

Authors:Alfredo Nájera Chávez Pages: 1007 - 1027 Abstract: 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)

Authors:Jan O. Kleppe; Rosa M. Miró-Roig Pages: 1029 - 1059 Abstract: 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)

Authors:Frederik Caenepeel; Fred Van Oystaeyen Abstract: Abstract We continue the study of glider representations of finite groups G with given structure chain of subgroups e ⊂ G 1 ⊂… ⊂ G d = G. We give a characterization of irreducible gliders of essential length e ≤ d which in the case of p-groups allows to prove some results about classical representation theory. The paper also contains an introduction to generalized character theory for glider representations and an extension of the decomposition groups in the Clifford theory. Furthermore, we study irreducible glider representations for products of groups and nilpotent groups. PubDate: 2017-07-31 DOI: 10.1007/s10468-017-9725-9

Authors:Hiroki Matsui Abstract: Abstract Classification problems of subcategories have been deeply considered so far. In this paper, we discuss classifying dense resolving and dense coresolving subcategories of exact categories via their Grothendieck groups. This study is motivated by the classification of dense triangulated subcategories of triangulated categories due to Thomason. PubDate: 2017-07-29 DOI: 10.1007/s10468-017-9726-8

Authors:Paul Balmer; Jon F. Carlson Abstract: Abstract We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module category and of the stable module category of any finite group. PubDate: 2017-07-26 DOI: 10.1007/s10468-017-9719-7

Authors:Shmuel Zelikson Abstract: Abstract Let \(\mathfrak {g}\) be a simple complex Lie algebra of types A n , D n , E n , and Q a quiver obtained by orienting its Dynkin diagram. Let λ be a dominant weight, and E(λ) the corresponding simple highest weight representation. We show that the weight multiplicities of E(λ) may be recovered by playing a numbers game Λ Q (λ), generalizing the well known Mozes game, constructing the orbit of λ under the action of the Weyl group W. The game board is provided by the Auslander-Reiten quiver Γ Q of Q. The game moves are obtained by constructing Nakajima’s monomial crystal M(λ) directly out of Γ Q . As an application, we consider Kashiwara’s parameterizations of the canonical basis. Let w 0 be a reduced expression of the longest element w 0 of W, adapted to a quiver Q of type A n . We show that a set of inequalities defining the string (Kashiwara) cone with respect to w 0, may be obtained by playing subgames of the numbers games Λ Q (ω i ) associated to fundamental representations. PubDate: 2017-07-25 DOI: 10.1007/s10468-017-9714-z

Authors:Charles F. Doran; Michael G. Faux; Sylvester J. Gates; Tristan Hübsch; Kevin Iga; Gregory D. Landweber Abstract: 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

Authors:Saeed Nasseh; Ryo Takahashi Abstract: 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

Authors:T. Geetha; Amritanshu Prasad Abstract: 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

Authors:John Hutchens; Nathaniel Schwartz Abstract: 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

Authors:Ralf Schiffler; Khrystyna Serhiyenko Abstract: 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