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:Sota Asai Abstract: Abstract We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable module categories have a structure of triangulated categories. Our main result determines the Grothendieck groups of these stable module categories. As an application, we give a complete classification of the mesh algebras up to stable equivalences. PubDate: 2017-09-07 DOI: 10.1007/s10468-017-9732-x

Authors:Ernie Stitzinger; Ashley White Abstract: Abstract We extend conjugacy results from Lie algebras to their Leibniz algebra generalizations. The proofs in the Lie case depend on anti-commutativity. Thus it is necessary to find other paths in the Leibniz case. Some of these results involve Cartan subalgebras. Our results can be used to extend other results on Cartan subalgebras. We show an example here and others will be shown in future work. PubDate: 2017-09-05 DOI: 10.1007/s10468-017-9731-y

Authors:Teresa Conde Abstract: Abstract The ADR algebra R A of an Artin algebra A is a right ultra strongly quasihereditary algebra (RUSQ algebra). In this paper we study the Δ-filtrations of modules over RUSQ algebras and determine the projective covers of a certain class of R A -modules. As an application, we give a counterexample to a claim by Auslander–Platzeck–Todorov, concerning projective resolutions over the ADR algebra. PubDate: 2017-09-02 DOI: 10.1007/s10468-017-9730-z

Authors:Yoshiyuki Kimura; Hironori Oya Abstract: Abstract In this paper, we show that quantum twist maps, introduced by Lenagan-Yakimov, induce bijections between dual canonical bases of quantum nilpotent subalgebras. As a corollary, we show the unitriangular property between dual canonical bases and Poincaré-Birkhoff-Witt type bases under the “reverse” lexicographic order. We also show that quantum twist maps induce bijections between certain unipotent quantum minors. PubDate: 2017-08-26 DOI: 10.1007/s10468-017-9729-5

Authors:Weideng Cui Abstract: Abstract In this paper, we prove that the pairwise orthogonal primitive idempotents of generic cyclotomic Birman-Murakami-Wenzl algebras can be constructed by consecutive evaluations of a certain rational function. In the Appendix, we prove a similar result for generic cyclotomic Nazarov-Wenzl algebras. A consequence of the constructions is a one-parameter family of fusion procedures for the cyclotomic Hecke algebra and its degenerate analogue. PubDate: 2017-08-23 DOI: 10.1007/s10468-017-9727-7

Authors:Ashish Gupta; Arnab Dey Sarkar Abstract: Abstract The Gelfand–Kirillov dimension has gained importance since its introduction as a tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand–Kirillov dimension of simple modules over certain simple rings of differential operators. We thus answer a question of J. C. McConnell in Representations of solvable Lie algebras V. On the Gelfand-Kirillov dimension of simple modules. McConnell (J. Algebra 76(2), 489–493, 1982) concerning this dimension for a class of algebras that arise as simple homomorphic images of solvable lie algebras. We also determine the Gelfand–Kirillov dimension of an induced module. PubDate: 2017-08-16 DOI: 10.1007/s10468-017-9728-6

Authors:Gena Puninski Abstract: Abstract We describe the Cohen–Macaulay part of the Ziegler spectrum and calculate Ringel’s quilt of the category of finitely generated Cohen–Macaulay modules over the A-infinity plane curve singularity. PubDate: 2017-08-07 DOI: 10.1007/s10468-017-9720-1

Authors:P. Petrullo; D. Senato Abstract: Abstract Starting from the invariant theory of binary forms, we extend the classical notion of covariants and introduce the ring of \(\mathcal {T}\) -covariants. This ring consists of maps defined on a ring of polynomials in one variable which commute with all the translation operators. We study this ring and we show some of its meaningful features. We state an analogue of the classical Hermite reciprocity law, and recover the Hilbert series associated with a suitable double grading via the elementary theory of partitions. Together with classical covariants of binary forms other remarkable mathematical notions, such as orthogonal polynomials and cumulants, turn out to have a natural and simple interpretation in this algebraic framework. As a consequence, a Heine integral representation for the cumulants of a random variable is obtained. PubDate: 2017-08-04 DOI: 10.1007/s10468-017-9724-x

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