Stephen Donkin, Haralampos Geranios
We consider the representation dimension, for fixed $n\geq2$, of ordinary and quantised Schur algebras $S(n,r)$ over a field $k$. For $k$ of positive characteristic $p$ we give a lower bound valid for all $p$. We also give an upper bound in the quantum case, when $k$ has characteristic $0$.
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Let $V$ be the two-dimensional simple module and $M$ be a projective Verma module for the quantum group of $\mathfrak{sl}_2$ at generic $q$. We show that for any $r\ge 1$, the endomorphism algebra of $M\otimes V^{\otimes r}$ is isomorphic to the type $B$ Temperley-Lieb algebra $\rm{TLB}_r(q, Q)$ for an appropriate parameter $Q$ depending on $M$. The parameter $Q$ is determined explicitly. We also use the cellular structure to determine precisely for which values of $r$ the endomorphism algebra is semisimple. A key element of our method is to identify the algebras $\rm{TLB}_r(q,Q)$ as the endom…
In this paper we prove Schur-Weyl duality between the symplectic group and Brauer algebra over an arbitrary infinite field $K$. We show that the natural homomorphism from the Brauer algebra $B_n(-2m)$ to the endomorphism algebra of tensor space $(K^{2m})^{\otimes n}$ as a module over the symplectic similitude group $GSp_{2m}(K)$ (or equivalently, as a module over the symplectic group $Sp_{2m}(K)$) is always surjective. Another surjectivity, that of the natural homomorphism from the group algebra for $GSp_{2m}(K)$ to the endomorphism algebra of $(K^{2m})^{\otimes n}$ as a module over $B_n(-2m)$…
It is shown that the endomorphism algebra of an arbitrary Young permutation module is cellular. Those are are quasi-hereditary are then determined.
On the Endomorphism Algebra of Specht Modules in Even Characteristic
本文针对特征2域上Specht模的自同态代数，提出了一种新颖描述，并给出了无限族具有一维自同态代数的Specht模块。
Let $A$ be a finite-dimensional algebra over a field of characteristic $p>0$. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective $A$--modules $P$ into those of the torsion submodules of $P$. As an application, we show that blocks of both the classical and quantum Schur algebras $S(2,r)$ and $S_q(2,r)$ are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain $2p^k$ simple modules for some $k$.
We prove that the stable endomorphism algebra of a module without self-extensions over a special biserial algebra is a gentle algebra. In particular, it is again special biserial. As a consequence, any algebra which is derived equivalent to a gentle algebra is gentle.
Geometric models for endomorphism algebras of tilting modules over gentle algebras
研究温和代数上倾斜模及其自同态代数，利用标记曲面与平铺框架，证明平铺忠实剖分对应倾斜模，定义辅助代数并证其与自同态代数同构，构造新平铺，引入保持翻转的倾斜模。
We find a relationship between the global dimension of an algebra $A$ and the global dimension of the endomorphism algebra of a $τ$-tilting module, when $A$ is of finite global dimension. We show that, in general, the global dimension of the endomorphism algebra is not always finite. For monomial algebras and special biserial algebras of global dimension two, we prove that the global dimension of the endomorphism algebra of any $τ$-tilting module is always finite. Moreover, for special biserial algebras, we give an explicit bound.
Let $k$ be a field of characteristic $2$ and let $H$ be a finite group or group scheme. We show that the negative Tate cohomology ring $\widehat{\text{H}}^{\leq 0}(H,k)$ can be realized as the endomorphism ring of the trivial module in a Verdier localization of the stable category of $kG$-modules for $G$ an extension of $H$. This means in some cases that the endomorphism of the trivial module is a local ring with infinitely generated radical with square zero. This stands in stark contrast to some known calculations in which the endomorphism ring of the trivial module is the degree zero compone…
Given a finite dimensional algebra $A$ over an algebraically closed field we study the relationship between the powers of the radical of a morphism in the module category of the algebra $A$ and the induced morphism in the module category of the endomorphism algebra of a tilting $A$-module. We compare the nilpotency indices of the radical of the mentioned module categories. We find an upper bound for the nilpotency index of the radical of the module category of iterated tilted algebras of Dynkin type.
Recently Tewari and van Willigenburg constructed modules of the 0-Hecke algebra that are mapped to the quasisymmetric Schur functions by the quasisymmetric characteristic and decomposed them into a direct sum of certain submodules. We show that these submodules are indecomposable by determining their endomorphism rings.
Building on recent work of Ardakov and Wadsley, we prove Schur's lemma for absolutely irreducible admissible p-adic Banach space (respectively locally analytic) representations of p-adic Lie groups. We also prove finiteness results for the endomorphism algebra of an irreducible admissible representation.
We show that for any singular dominant integral weight $λ$ of a complex semisimple Lie algebra $\mathfrak{g}$, the endomorphism algebra $B$ of any projective-injective module of the parabolic BGG category $\mathcal{O}_λ^{\mathfrak{p}}$ is a symmetric algebra (as conjectured by Khovanov) extending the results of Mazorchuk and Stroppel for the regular dominant integral weight. Moreover, the endomorphism algebra $B$ is equipped with a homogeneous (non-degenerate) symmetrizing form. In the appendix, there is a short proof due to K. Coulembier and V. Mazorchuk showing that the endomorphism algebra …
Suppose that $G$ is a finite group and $k$ is a field of characteristic $p >0$. Let $\mathcal{M}$ be the thick tensor ideal of finitely generated modules whose support variety is in a fixed subvariety $V$ of the projectivized prime ideal spectrum $\operatorname{Proj} \operatorname{H}^*(G,k)$. Let $\mathcal{C}$ denote the Verdier localization of the stable module category $\operatorname{stmod}(kG)$ at $\mathcal{M}$. We show that if $V$ is a finite collection of closed points and if the $p$-rank every maximal elementary abelian $p$-subgroups of $G$ is at least 3, then the endomorphism ring of th…
Let $Σ_r$ be the symmetric group acting on $r$ letters, $K$ be a field of characteristic 2 and $λ$ and $μ$ be partitions of $r$ in at most two parts. Denote the permutation module corresponding to the Young subgroup $Σ_λ$ in $Σ_r$ by $M^λ$, and the indecomposable Young module by $Y^μ$. We give an explicit presentation of the endomorphism algebra ${\rm End}_{K[Σ_r]}(Y^μ)$, using the idempotents found by Doty, Erdmann and Henke in [1].
We study the endomorphism algebra of Verma modules for rational Cherednik algebras at t=0. It is shown that, in many cases, these endomorphism algebras are quotients of the centre of the rational Cherednik algebra. Geometrically, they define Lagrangian subvariaties of the generalized Calogero-Moser space. In the introduction, we motivate our results by describing them in the context of derived intersections of Lagrangians.
A note on Schur-Weyl dualities for $GL(m)$ and $GL(m|n)$
本文用统一的基础方法证明了任意特征无限基域上一般线性群和超群的经典、混合、超及混合超 Schur-Weyl 对偶性的第二部分，主要新结果是正特征情形下的混合及混合超对偶性。
In this paper, we first show that for an acyclic gentle algebra A, the irreducible components of any moduli space of A-modules are products of projective spaces. Next, we show that the nice geometry of the moduli spaces of modules of an algebra does not imply the tameness of the representation type of the algebra in question. Finally, we place these results in the general context of moduli spaces of modules of Schur-tame algebras. More specifically, we show that for an arbitrary Schur-tame algebra A and theta-stable irreducible component C of a module variety of A-modules, the moduli space of …
Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. We investigate the structure properties of the endomorphism algebras of semi-tilting $A$-modules, and prove that the endomorphism algebras arising from the mutations of semi-tilting $A$-modules can be realized as the endomorphism algebras of BB-tilting modules.
Given $r \in \mathbf{N},$ let $λ$ be a partition of $r$ with at most two parts. Let $\mathbf{F}$ be a field of characteristic 3. Write $M^λ$ for the $\mathbf{F}S_r$-permutation module corresponding to the action of the symmetric group $S_r$ on the cosets of the maximal Young subgroup $S_λ.$ We construct a full set of central primitive idempotents in $\text{End}_{\mathbf{F} S_r}(M^λ)$ in this case. We also determine the Young module corresponding to each primitive idempotent that we construct.
Let $W$ be a finite Weyl group and $\sg$ be a non-trivial graph automorphism of $W$. We show a remarkable relation between the $\sg$-twisted involution module for $W$ and the Frobenius--Schur indicators of the unipotent characters of a corresponding twisted finite group of Lie type. This extends earlier results of Lusztig-Vogan for the untwisted case and then allows us to state a general result valid for any finite group of Lie type. Inspired by recent work of Marberg, we also formally define Frobenius--Schur indicators for "unipotent characters" of twisted dihedral groups.
We determine the partitions $λ$ for which the corresponding induced module (or Schur module in the language of Buchsbaum et. al., [1]) $\nabla(λ)$ is injective in the category of polynomial modules for a general linear group over an infinite field, equivalently which Weyl modules are projective polynomial modules. Since the problem is essentially no more difficult in the quantised case we address it at this level of generality. Expressing our results in terms of the representation theory of Hecke algebras at the parameter $q$ we determine the partitions $λ$ for which the corresponding Specht m…
Let $A$ be an Artin algebra, $M$ be a Gorenstein projective $A$-module and $B =$ End$_A M$, then $M$ is a $A$-$B$-bimodule. We use the restricted flat dimension of $M_B$ to give a characterization of the homological dimensions of $A$ and $B$, and obtain the following main results: (1) if $A$ is a CM-finite algebra with $\cal GP$($A$) = add$_AE$ and fin.dim $A \geq 2,$ then ${\rm fin.dim}\ B \leq {\rm fin.dim}\ A + {\rm rfd}(M_B) +{\rm pd}_B{\rm Hom}_A(M, E);$ (2) If $A$ is a CM-finite $n$-Gorenstein algebra with $\cal GP$($A$) = add$_AE$ and $n \geq 2$, then gl.dim $B \leq n + {\rm pd}_B{\rm H…
The extended Schur functions form a basis of quasisymmetric functions that contains the Schur functions. We provide a representation-theoretic interpretation of this basis by constructing $0$-Hecke modules whose quasisymmetric characteristics are the extended Schur functions. We further prove these modules are indecomposable.
Let $C$ be a symmetrizable generalized Cartan matrix with symmetrizer $D$ and orientation $Ω$. In previous work we associated an algebra $H$ to this data, such that the locally free $H$-modules behave in many aspects like representations of a hereditary algebra $\tilde{H}$ of the corresponding type. We define a Noetherian algebra $\hat{H}$ over a power series ring, which provides a direct link between the representation theory of $H$ and of $\tilde{H}$. We define and study a reduction and a localization functor relating the module categories of these three types of algebras. These are used to …
Given a quasi-hereditary superalgebra $A$, the first author and R. Muth have defined generalized Schur bi-superalgebras $T^A(n)$ and proved that these algebras are again quasi-hereditary. In particular, $T^A(n)$ comes with a family of standard modules. Developing the work of Donkin and Mathieu on good filtrations, we prove that tensor product of standard modules over $T^A(n)$ has a standard filtration.
Flagged Schur modules generalize the irreducible representations of the general linear group under the action of the Borel subalgebra. Their characters include many important generalizations of Schur polynomials, such as Demazure characters, flagged skew Schur polynomials, and Schubert polynomials. In this paper, we prove the characters of flagged Schur modules can be computed using a simple combinatorial algorithm due to Kohnert if and only if the indexing diagram is northwest. This gives a new proof that characters of flagged Schur modules are nonnegative sums of Demazure characters and give…
We present an independent short proof of the main result of arXiv:0706.3725 that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. We derive this from the results of arXiv:0712.1183 about the shift of argument subalgebras.
In this paper we consider the (affine) Schur algebra introduced by Vignéras as the endomorphism algebra of certain permutation modules for the Iwahori-Matsumoto Hecke algebra. This algebra describes, for a general linear group over a $p$-adic field, a large part of the unipotent block over fields of characteristic different from $p$. We show that this Schur algebra is, after a suitable completion, isomorphic to the quiver Schur algebra attached to the cyclic quiver. The isomorphism is explicit, but nontrivial. As a consequence, the completed (affine) Schur algebra inherits a grading. As a bypr…
In this note we give defining relations of an $\mathfrak{sl}_{n+1}[t]$-module defined by the fusion product of simple $\mathfrak{sl}_{n+1}$-modules whose highest weights are multiples of a given fundamental weight. From this result we obtain a surjective homomorphism between two fusion products, which can be considered as a current algebra analog of Schur positivity.
Asymptotic Schur algebras and cellularity of q-Schur algebras
本文证明了有限型 q-Schur 代数的胞腔性，推广了 Geck 的定理，并研究了渐近 Schur 代数的特殊模与 Schur 代数的左胞腔表示，推广了 Lusztig 的相关工作。
Let $K$ be a field of characteristic two, and let $λ$ be a two-part partition of some natural number $r$. Denote the permutation module corresponding to the (maximal) Young subgroup $Σ_λ$ in $Σ_r$ by $M^λ$. We construct a full set of orthogonal primitive idempotents of the centraliser subalgebra $S_K(λ) = 1_λS_K(2,r) 1_λ= End_{KΣ_r}(M^λ)$ of the Schur algebra $S_K(2,r)$. These idempotents are naturally in one-to-one correspondence with the 2-Kostka numbers.
Yangians and degenerate affine Schur algebras
本文通过图示计算法描述了Yangian到退化有理Schur代数的同态，并在特定条件下计算了其核，从而给出了该代数的表示。
In this paper, we construct the $q$-Schur modules as left principle ideals of the cyclotomic $q$-Schur algebras, and prove that they are isomorphic to those cell modules defined in \cite{8} and \cite{15} at any level $r$. Then we prove that these $q$-Schur modules are free modules and construct their bases. This result gives us new versions of several results about the standard basis and the branching theorem. With the help of such realizations and the new bases, we re-prove the Branch rule of Weyl modules which was first discovered and proved by Wada in \cite{23}.
We construct modules of the $0$-Hecke algebra whose images under the quasisymmetric characteristic map are the Young row-strict quasisymmetric Schur functions. This provides a representation-theoretic interpretation of this basis of quasisymmetric functions, answering a question of Mason and Niese (2015). Additionally, we classify when these modules are indecomposable.
We investigate the endotrivial modules for the Schur covers of the symmetric and alternating groups and determine the structure of their group of endotrivial modules in all characteristics. We provide a full description of this group by generators and relations in all cases.
Let G be GL_N or SL_N as reductive linear algebraic group over a field k of positive characteristic p. We prove several results that were previously established only when N < 6 or p > 2^N. Let G act rationally on a finitely generated commutative k-algebra A. Assume that A as a G-module has a good filtration or a Schur filtration. Let M be a noetherian A-module with compatible G action. Then M has finite good/Schur filtration dimension, so that there are at most finitely many nonzero H^i(G,M). Moreover these H^i(G,M) are noetherian modules over the ring of invariants A^G. Our main tool is a res…
On Noetherian algebras, Schur functors and Hemmer-Nakano dimensions
本文研究（投射）诺特代数B的模范畴与B上生成-相对上生成子的自同态代数的模范畴间联系的质量，通过舒尔函子分析Hemmer-Nakano维数，建立整数情形与有限维情形的计算关联，推导舒尔代数和BGG范畴O的结果并展示形变的结构性质。
Let $A$ be a standardly stratified algebra over a field $K$ and $T$ a tilting module over $A$. Let $Λ^+$ be an indexing set of all simple modules in $A\lmod$. We show that if there is an integer $r\in\N$ such that for any $λ\inΛ^+$, there is an embedding $Δ(λ)\hookrightarrow T^{\oplus r}$ as well as an epimorphism $T^{\oplus r}\twoheadrightarrow\overline{\nabla}(λ)$ as $A$-modules, then $T$ is a faithful $A$-module and $A$ has the double centraliser property with respect to $T$. As applications, we prove that if $A$ is quasi-hereditary with a simple preserving duality and $T$ a given faithful …
A new basis for the polynomial ring of infinitely many variables is constructed which consists of products of Schur functions and Q-functions. The transition matrix from the natural Schur function basis is investigated.
The image of the Specht module under the inverse Schur functor in arbitrary characteristic
本文研究了任意特征下 Specht 模在逆 Schur 函子下的像，给出了特征 2 下与对偶 Weyl 模同构的充要条件，并在其他特征下给出了初等证明，同时探讨了同构不成立时的商映射性质。
We prove that there is a natural grading-preserving isomorphism of $\sl$-modules between the basic module of the affine Lie algebra $\widehat\sl$ (with the homogeneous grading) and a Schur--Weyl module of the infinite symmetric group $\sinf$ with a grading defined through the combinatorial notion of the major index of a Young tableau, and study the properties of this isomorphism. The results reveal new and deep interrelations between the representation theory of $\widehat\sl$ and the Virasoro algebra on the one hand, and the representation theory of $\sinf$ and the related combinatorics on the…
Endomorphism algebras of silting complexes
本文研究遗传代数导出范畴中$n$项倾斜复形的自同态代数，证明其模范畴具有分离的$n$截面，并特例给出三分割。
Schur modules give the irreducible polynomial representations of the general linear group $\mathrm{GL}_t$. Viewing the symmetric group $\mathfrak{S}_t$ as a subgroup of $\mathrm{GL}_t$, we may restrict Schur modules to $\mathfrak{S}_t$ and decompose the result into a direct sum of Specht modules, the irreducible representations of $\mathfrak{S}_t$. We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for $\mathfrak{S}_t$ for $t$ large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternat…
Let R be a Cohen-Macaulay ring and M a maximal Cohen-Macaulay R-module. Inspired by recent striking work by Iyama, Burban-Iyama-Keller-Reiten and Van den Bergh we study the question of when the endomorphism ring of M has finite global dimension via certain conditions about vanishing of $\Ext$ modules. We are able to strengthen certain results by Iyama on connections between a higher dimension version of Auslander correspondence and existence of non-commutative crepant resolutions. We also recover and extend to positive characteristics a recent Theorem by Burban-Iyama-Keller-Reiten on cluster-t…
In this paper, we define the Yokonuma-Schur algebra $\text{YS}_{q}(r,n)$ as the endomorphism algebra of a permutation module for the Yokonuma-Hecke algebra $\text{Y}_{r,n}(q).$ We prove that $\text{YS}_{q}(r,n)$ is cellular by constructing an explicit cellular basis following the approach in [DJM], and we further show that it is a quasi-hereditary cover of $\text{Y}_{r,n}(q)$ in the sense of Rouquier following [HM2]. We also introduce the tilting modules for $\text{YS}_{q}(r,n).$ In the appendix, we define and study the cyclotomic Yokonuma-Schur algebra in a similar way.
We show that every two-term tilting complex over an Artin algebra has a tilting module over a certain factor algebra as a homology group. Also, we determine the endomorphism algebra of such a homology group, which is given as a certain factor algebra of the endomorphism algebra of the two-term tilting complex. Thus, every derived equivalence between Artin algebras given by a two-term tilting complex induces a derived equivalence between the corresponding factor algebras.
We compute explicitly the submodule structure of the Young modules for symmetric groups $S_n$ over fields of characteristic 2, when $n \le 7$. We use this information to compute the submodule structure of indecomposable projectives for the corresponding Schur algebras when $n \le 5$, and we give give partial information when n=6,7, including the Gabriel quiver and the structure of the Weyl modules. We resolve all Morita equivalences between blocks of these algebras.
Motivated by recent problems regarding the symmetry of Hecke algebras, we investigate the symmetry of the endomorphism algebra $E_P(M)$ for $P$ a $p$-group and $M$ a $kP$-module with $k$ a field of characteristic $p$. We provide a complete analysis for cyclic $p$-groups and the dihedral 2-groups. For the dihedral 2-groups, this requires the classification of the indecomposable modules in terms of string modules and band modules. We generalize our techniques to consider $E_Λ(M)$ for $Λ$ a Nakayama algebra, a local algebra, or even an arbitrary algebra.
Representation Type and Schurian Finiteness of Tits Algebras
该研究确定了代数闭域上有限中心实超平面排列的蒂茨代数的表示类型与舒尔有限性，给出了不同超平面数量对应的分类结果，并通过多种代数方法完成证明，还构造了相关模的嵌入。
We study on Weyl modules of cyclotomic $q$-Schur algebras. In particular, we give the character formula of the Weyl modules by using the Kostka numbers and some numbers which are computed by a generalization of Littlewood-Richardson rule. We also study corresponding symmetric functions. Finally, we give some simple applications to modular representations of cyclotomic $q$-Schur algebras.
We study imaginary representations of the Khovanov-Lauda-Rouquier algebras of affine Lie type. Irreducible modules for such algebras arise as simple heads of standard modules. In order to define standard modules one needs to have a cuspidal system for a fixed convex preorder. A cuspidal system consists of irreducible cuspidal modules---one for each real positive root for the corresponding affine root system ${\tt X}_l^{(1)}$, as well as irreducible imaginary modules---one for each $l$-multipartition. We study imaginary modules by means of `imaginary Schur-Weyl duality'. We introduce an imagina…
When the parameter $q\in\mathbb C^*$ is not a root of unity, simple modules of affine $q$-Schur algebras have been classified in terms of Frenkel--Mukhin's dominant Drinfeld polynomials (\cite[4.6.8]{DDF}). We compute these Drinfeld polynomials associated with the simple modules of an affine $q$-Schur algebra which come from the simple modules of the corresponding $q$-Schur algebra via the evaluation maps.
The affine Schur algebra $\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of type $A_{r-1}$. By the affine Schur-Weyl duality it is isomorphic to the image of the representation map of the $\mathcal{U}(\hat{\mathfrak{gl}}_{n})$ action on the tensor space when $K$ is the field of complex numbers. We show that $\widetilde{S}(n,r)$ can be defined in another two equivalent ways. Namely, it is the image of the representation map of the semigroup algebra $K\widetilde{GL}_{n,a}$ (defined in Section \ref…
We exhibit for all positive integers r, an explicit cellular structure for the endomorphism algebra of the r'th tensor power of an integral form of the Weyl module with highest weight d of the quantised enveloping algebra of sl2. When q is specialised to a root of unity of order bigger than d, we consider the corresponding specialisation of the tensor power. We prove one general result which gives sufficient conditions for the commutativity of specialisation with the taking of endomorphism algebras, and another which relates the multiplicities of indecomposable summands to the dimensions of si…
A non-iterative straightening algorithm and orthogonality for skew Schur modules
本文将 Fulton 的 Schur 模行列式构造推广至斜设置，提出了一种非迭代拉直算法以显式闭系数公式将任意元素表示为新的 D-基，并证明该基源于对半标准表基的 Gram-Schmidt 正交化，从而揭示了自然内积并将拉直重表述为正交投影。
As a natural generalization quantum Schur algebras associated with the Hecke algebra of the symmetric group, we introduce the quantum Schur superalgebra of type Q associated with the Hecke-Clifford superalgebra, which, by definition, is the endomorphism algebra of the induced module over the Hecke-Clifford superalgebra from certain permutation modules over the Hecke algebra of the symmetric group. We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base change property for them. We will also identify the quantum Schur superalgebra of type…
This paper investigates the tilting modules of the cyclotomic q-Schur algebras, the Young modules of the Ariki-Koike algebras, and the interconnections between them. The main tools used to understand the tilting modules are contragredient duality, and the Specht filtrations and dual Specht filtrations of certain permutation modules. Surprisingly, Weyl filtrations -- which are in general more powerful than Specht filtrations -- play only a secondary role.
In this note, inspired by the proof of the Kirillov-Reshetikhin conjecture, we consider tensor products of Kirillov-Reshetikhin modules of a fixed node and various level. We fix a positive integer and attach to each of its partitions such a tensor product. We show that there exists an embedding of the tensor products, with respect to the classical structure, along with the reverse dominance relation on the set of partitions.
We study the structure of the $0$-Schur algebra $S_0(n, r)$ following the geometric construction of $S_0(n, r)$ by Jensen and Su \cite{JS}. The main results are the construction and classification of indecomposable projective modules. In addition, we construct bases of these modules and their homomorphism spaces. We also give a filtration of projective modules, which leads to a decomposition of $S_0(n,r)$ into indecomposable left modules.
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