Let $α$ be a composition of $n$ and $σ$ a permutation in $\mathfrak{S}_{\ell(α)}$. This paper concerns the projective covers of $H_n(0)$-modules $\mathcal{V}_α$, $X_α$ and $\mathbf{S}^σ_α$, which categorify the dual immaculate quasisymmetric function, the extended Schur function, and the quasisymmetric Schur function when $σ$ is the identity, respectively. First, we show that the projective cover of $\mathcal{V}_α$ is the projective indecomposable module $\mathbf{P}_α$ due to Norton, and $X_α$ and the $ϕ$-twist of the canonical submodule $\mathbf{S}^σ_{β,C}$ of $\mathbf{S}^σ_β$ for $(β,σ)$'s satisfying suitable conditions appear as $H_n(0)$-homomorphic images of $\mathcal{V}_α$. Second, we introduce a combinatorial model for the $ϕ$-twist of $\mathbf{S}^σ_α$ and derive a series of surjections starting from $\mathbf{P}_α$ to the $ϕ$-twist of $\mathbf{S}^{\mathrm{id}}_{α,C}$. Finally, we construct the projective cover of every indecomposable direct summand $\mathbf{S}^σ_{α, E}$ of $\mathbf{S}^σ_α$. As a byproduct, we give a characterization of triples $(σ, α, E)$ such that the projective cover of $\mathbf{S}^σ_{α, E}$ is indecomposable.
We study the $H_n(0)$-module $\mathbf{S}^σ_α$ due to Tewari and van Willigenburg, which was constructed using new combinatorial objects called standard permuted composition tableaux and decomposed into cyclic submodules. First, we show that every direct summand appearing in their decomposition is indecomposable and characterize when $\mathbf{S}^σ_α$ is indecomposable. Second, we find characteristic relations among $\mathbf{S}^σ_α$'s and expand the image of $\mathbf{S}^σ_α$ under the quasi characteristic in terms of quasisymmetric Schur functions. Finally, we show that the canonical submodule o…
We study the torsion free generalized crystallographic groups with the indecomposable holonomy group which is isomorphic to either a cyclic group of order ${p^s}$ or a direct product of two cyclic groups of order ${p}$.
This article gives the structure of all projective indecomposable modules in blocks with non-cyclic defect group for $n\leq 11$, and almost all in the case of $n=12$, leaving four simple modules for $p=2$ and one simple module for $p=3$.
We begin by deriving an action of the 0-Hecke algebra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristics are the natural refinements of Schur functions known as quasisymmetric Schur functions. Furthermore, we classify combinatorially which of these 0-Hecke modules are indecomposable. From here, we establish that the natural equivalence relation arising from our 0-Hecke action has equivalence classes that are isomorphic to subintervals of the weak Bruhat order on the symmetric group. Focussing on the equivalence classes containin…
Relying on the classification of the indecomposable liftable modules in arbitrary blocks with non-trivial cyclic defect groups we give a complete classification of the trivial source modules lying in such blocks, describing in particular their associated path on the Brauer tree of the block in the sense of Janusz (1969). The appendix contains a description of the minimal distance from an arbitrary non-projective indecomposable liftable module to the boundary of the stable Auslander-Reiten quiver of the block.
In this paper, we characterize Ulrich modules over cyclic quotient surface singularities by using the notion of special Cohen-Macaulay modules. We also investigate the number of indecomposable Ulrich modules for a given cyclic quotient surface singularity, and show that the number of exceptional curves in the minimal resolution gives a boundary for the number of indecomposable Ulrich modules.
Let $G$ be a cyclic group of order $p$, let $k$ be a field of characteristic $p$, and let $V, W$ be $kG$-modules. We study the modules of covariants $k[V,W]^G = (S(V^*) \otimes W)^G$. For $V$ indecomposable with dimension 2, and $W$ an arbitrary indecomposable module, we show $k[V,W]^G$ is a free $k[V]^G$-module (recovering a result of Broer and Chuai) and we give an explicit set of covariants generating $k[V,W]^G$ freely over $k[V]^G$. For $V$ indecomposable with dimension 3 and $W$ an indecomposable module with dimension at most 5, we show that $k[V,W]^G$ is a Cohen-Macaulay $k[V]^G$-module …
Let $F$ be a field of non-zero characteristic $p$, let $G$ be a cyclic group of order $q =p^a$ for some positive integer $a$, and let $U$ and $W$ be indecomposable $F G$-modules. We identify a generator for each of the indecomposable components of $U \otimes W$ in terms of a particular $F$-basis of $U \otimes W$, extending previous work in which each component occurred with multiplicity one. An anonymous reviewer helped us realize that our results are really just a translation of previous results of Iiwa and Iwamatsu to our situation.
We introduce a general constructive method to find a p-basis (and the Ulm invariants) of a finite Abelian p-group M. This algorithm is based on Groebner bases theory. We apply this method to determine the additive structure of indecomposable modules over the following Dedeking-like rings: ZCp, where Cp is the cyclic group of order a prime p, and the p-pullback Z --> Zp <-- Z of Z+Z.
Let $A$ be the path algebra of a Dynkin quiver $Q$ over a finite field, and $\mathscr{P}$ be the category of projective $A$-modules. Denote by $C^1(\mathscr{P})$ the category of 1-cyclic complexes over $\mathscr{P}$, and $\tilde{\mathfrak{n}}^+$ the vector space spanned by the isomorphism classes of indecomposable and non-acyclic objects in $C^1(\mathscr{P})$. In this paper, we prove the existence of Hall polynomials in $C^1(\mathscr{P})$, and then establish a relationship between the Hall numbers for indecomposable objects therein and those for $A$-modules. Using Hall polynomials evaluated at…
Suppose that $Q$ is a connected quiver without oriented cycles and $σ$ is an automorphism of $Q$. Let $k$ be an algebraically closed field whose characteristic does not divide the order of the cyclic group $\langleσ\rangle$. The aim of this paper is to investigate the relationship between indecomposable $kQ$-modules and indecomposable $kQ\#k\langleσ\rangle$-modules. It has been shown by Hubery that any $kQ\#k\langleσ\rangle$-module is an isomorphically invariant $kQ$-module, i.e., ii-module (in this paper, we call it $\langleσ\rangle$-equivalent $kQ$-module), and conversely any $\langleσ\rangl…
The purpose of this paper is to provide a unified method for dealing with various 0-Hecke modules constructed using tableaux so far. To do this, we assign a $0$-Hecke module to each left weak Bruhat interval, called a weak Bruhat interval module. We prove that every indecomposable summand of the $0$-Hecke modules categorifying dual immaculate quasisymmetric functions, extended Schur functions, quasisymmetric Schur functions, and Young row-strict quasisymmetric Schur functions is a weak Bruhat interval module. We further study embedding into the regular representation, induction product, restri…
If $K$ is a field of finite characteristic $p$, $G$ is a cyclic group of order $q=p^α$, $U$ and $W$ are indecomposable $KG$-modules with $\dim U=m$ and $\dim W=n$, and $λ(m,n,p)$ is a standard Jordan partition of $ m n$, we describe how to find a generator for each of the indecomposable components of the $KG$-module $U \otimes W$.
In this paper we classify all the cyclic finite dimensional indecomposable\\ modules of the perfect Lie algebras $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$, given by the semidirect sum of the simple Lie algebra $A_n$ with its standard representation. Furthermore, using the embeddings of the Lie algebras $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$ in $\mathfrak{sl}(n+2)$, we show that any finite dimensional irreducible module of $\mathfrak{sl}(n+2)$ restricted to $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$ is a cyclic module and that any cyclic $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1…
Let K be a principal ideal domain, G a finite group, and M a KG-module which as K-module is free of finite rank, and on which $G$ acts faithfully. A generalized crystallographic group (introduced by the authors in volume 5 of Journal of Group Theory) is a group $\frak C$ which has a normal subgroup isomorphic to M with quotient G, such that conjugation in $\frak C$ gives the same action of G on M that we started with. (When $K=\Bbb Z$, these are just the classical crystallographic groups.) The K-free rank of M is said to be the dimension of $\frak C$, the holonomy group of $\frak C$ is G, and …
A ring $R$ is called left $k$-cyclic if every left $R$-module is a direct sum of indecomposable modules which are homomorphic image of $_{R}R^k$. In this paper, we give a characterization of left $k$-cyclic rings. As a consequence, we give a characterization of left Köthe rings, which is a generalization of Köthe-Cohen-Kaplansky theorem. We also characterize rings which are Morita equivalent to a basic left $k$-cyclic ring. As a corollary, we show that $R$ is Morita equivalent to a basic left Köthe ring if and only if $R$ is an artinian left multiplicity-free top ring.
We study the classical Köthe's problem, concerning the structure of non-commutative rings with the property that: ``every left module is a direct sum of cyclic modules". In 1934, Köthe showed that left modules over Artinian principal ideal rings are direct sums of cyclic modules. A ring $R$ is called a ${\it left~Köthe~ring}$ if every left $R$-module is a direct sum of cyclic $R$-modules. In 1951, Cohen and Kaplansky proved that all commutative K{ö}the rings are Artinian principal ideal rings. During the years 1962 to 1965, Kawada solved the Köthe's problem for basic fnite-dimensional algebras…
Let $n$ be a nonnegative integer. For each composition $α$ of $n$, Berg $\textit{et al.}$ introduced a cyclic indecomposable $H_n(0)$-module $\mathcal{V}_α$ with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study $\mathcal{V}_α$'s from the homological viewpoint. To be precise, we construct a minimal projective presentation of $\mathcal{V}_α$ and a minimal injective presentation of $\mathcal{V}_α$ as well. Using them, we compute ${\rm Ext}^1_{H_n(0)}(\mathcal{V}_α, {\bf F}_β)$ and ${\rm Ext}^1_{H_n(0)}( {\bf F}_β, \mathcal{V}_α)$…
Let E be a cyclic extension of degree p^n of a field F of characteristic p. Using arithmetic invariants of E/F we determine k_mE, the Milnor K-groups K_mE modulo p, as Fp[Gal(E/F)]-modules for all m in N. In particular, we show that each indecomposable summand of k_mE has Fp-dimension a power of p. That all powers p^i, i=0,1,...,n, occur for suitable examples is shown in a subsequent paper [MSS2], where additionally the main result of this paper becomes an essential induction step in the determination of K_mE/p^sK_mE as (Z/p^sZ)[Gal(E/F)]-modules for all m, s in N.
