Philippe Caldero, Peter Littelmann
Let G be a complex semisimple Lie group. The aim of this article is to compare two basis for G-modules, namely the standard monomial basis and the dual canonical basis. In particular, we give a sufficient condition for a standard monomial to be an element of the dual canonical basis and vice versa.
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We establish a link between the Gerstenhaber bracket in the Hochshild cohomology and the behaviour of cycles in the module category of a monomial quadratic algebra $A$.
We give a new realization of crystal bases for finite dimensional irreducible modules over special linear Lie algebras using the monomials introduced by H. Nakajima. We also discuss the connection between this monomial realization and the tableau realization given by Kashiwara and Nakashima.
Let $A$ be a finite dimensional algebra having the double centraliser property with respect to a minimal faithful projective-injective left module $Af$ for some idempotent $f$. We prove that in this case $A$ is a monomial algebra if and only if $A$ is a Nakayama algebra given by quiver and relations.
A celebrated result by Keller--Reiten says that $2$-Calabi--Yau tilted algebras are Gorenstein and stably $3$-Calabi--Yau. This note shows that the converse holds in the monomial case: a $1$-Gorenstein monomial algebra with a $3$-Calabi--Yau singularity category is $2$-Calabi--Yau tilted. We study the case of other Goresntein monomial algebras with stably Calabi--Yau singularity categories.
The Auslander-Gorenstein condition for monomial algebras
本文研究了单项代数的 Auslander-Gorenstein 性质，通过将其归结为 Nakayama algebra 的分类并建立与双射 Auslander-Reiten 映射的等价关系，给出了新的同调刻画并证实了相关猜想。
In this paper, we consider the singularity category $D_{sg}(\mod A)$ and the $\mathbb{Z}$-graded singularity category $D_{sg}(\mod^{\mathbb Z} A)$ for a Gorenstein monomial algebra $A$. Firstly, for a positively graded $1$-Gorenstein algebra, we prove that its ${\mathbb Z}$-graded singularity category admits silting objects. Secondly, for $A=KQ/I$ being a Gorenstein monomial algebra, we prove that $D_{sg}(\mod^{\mathbb Z} A)$ has tilting objects. As a consequence, $D_{sg}(\mod^{\mathbb Z}A)$ is triangulated equivalent to the derived category of a hereditary algebra $H$ which is of finite repre…
The Hochschild cohomology ring of monomial algebras
本文为单项式代数构造Bardzell分解上的显式对角映射，用以描述杯积，并证明三角单项式代数正次数杯积为零，方法基于分次交换性而非显式计算。
We study the Lie module structure given by the Gerstenhaber bracket on the Hochschild cohomology groups of a monomial algebra with radical square zero. The description of such Lie module structure will be given in terms of the combinatorics of the quiver. The Lie module structure will be related to the classification of finite dimensional modules over simple Lie algebras when the quiver is given by the two loops and the ground field is the complex numbers.
Let A be a basic connected finite dimensional algebra over an algebraically closed field k. Assuming that A is monomial and that the ordinary quiver Q of A has no oriented cycle and no multiple arrows, we prove that A admits a universal cover with group the fundamental group of the underlying space of Q.
We use directed graphs called "syzygy quivers" to study the asymptotic growth rates of the dimensions of the syzygies of representations of finite dimensional algebras. For any finitely generated representation of a monomial algebra, we show that this growth rate is poly-exponential, i.e. the product of a polynomial and an exponential function, and give a procedure for computing the corresponding degree and base from a syzygy quiver. We characterize the growth rates arising in this context: The bases of the occurring exponential functions are the real, nonnegative algebraic integers $b$ whose …
Let $\mathbf{k}$ be a field of arbitrary characteristic, let $Λ$ be a finite dimensional $\mathbf{k}$-algebra, and let $V$ be an indecomposable Gorenstein-projective $Λ$-module with finite dimension over $\mathbf{k}$. It follows that $V$ has a well-defined versal deformation ring $R(Λ, V)$, which is complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$, and which is universal provided that the stable endomorphism ring of $V$ is isomorphic to $\mathbf{k}$. We prove that if $Λ$ is a monomial algebra without overlaps, then $R(Λ,V)$ is universal and isomorphic …
Classification of Auslander-Gorenstein monomial algebras: The acyclic case
本文通过Coxeter矩阵的Bruhat分解对无环单则代数进行线性代数分类，证明了在温和假设下，单则无环有向图代数为Auslander正则当且仅当其Coxeter矩阵有Bruhat分解。
In this paper we give a direct proof of the positivity conjecture for adapted quantum cluster variables. Moreover, our process allows one to explicitly compute formulas for all adapted cluster monomials and certain ordered products of adapted cluster monomials. In particular, we describe all cluster monomials in cluster algebras and quantum cluster algebras of rank 2. One may obtain similar formulas for all finite type cluster monomials. The above results are achieved by computing explicit set-theoretic decompositions of Grassmannians of subrepresentations in adapted valued quiver representati…
Exact Borel subalgebras of quasi-hereditary monomial algebras
本文研究了单项拟遗传代数在König意义下何时存在精确Borel子代数，证明其始终具有由精确Borel子代数和对偶子代数组成的Reedy分解，并给出了该子代数的显式描述、唯一性及正则性判据。
We classify indecomposable non-projective Gorenstein-projective modules over a monomial algebra via the notion of perfect paths. We apply this classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.
Let $K$ be a field of characteristic 0 containing all roots of unity. We classify all the Hopf structures on monomial $K$-coalgebras, or, in dual version, on monomial $K$-algebras.
Koszul Duality for Quadratic Monomial Algebras
研究有限维二次单项式代数及其Koszul对偶的模结构，证明Koszul对偶是分级连贯且分级余连贯的，并推导出绝对Koszul性、全局线性缺陷至多为1等性质，进而细化导出Koszul对偶和Bernstein-Gelfand-Gelfand对应。
We present explicit descriptions of the crystals $\mathcal{B}(\infty)$ and $\mathcal{B}(λ)$ over special linear Lie algebras in the language of \emph{extended Nakajima monomials}. There is a natural correspondence between the monomial description and Young tableau realization, which is another realization of crystals $\mathcal{B}(\infty)$ and $\mathcal{B}(λ)$.
A note on injectivity of monomial algebras
本文研究单项式代数的自内射性，通过正则模的 socle 映射可扩张条件给出等价刻画与完整分类，并证明此类代数均属于 Nakayama 代数。
We present a vertex operator algebra which is an extension of the level $k$ vertex operator algebra for the $\hat{sl}_2$ conformal field theory. We construct monomial basis of its irreducible representations.
We study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra A, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of A in characteristic zero. Criteria for the (semi-)simplicity, the solvability, the reductivity, the commutativity and the nilpotency are given.
In this paper we give necessary and sufficient conditions for the variety of a simple module over a (D,A)-stacked monomial algebra to be nontrivial. This class of algebras was introduced in [Green and Snashall, The Hochschild cohomology ring modulo nilpotence of a stacked monomial algebra, Colloq. Math. 105 (2006), 233-258] and generalizes Koszul and D-Koszul monomial algebras. As a consequence we show that if the variety of every simple module over such an algebra is nontrivial then the algebra is D-Koszul. We give examples of (D,A)-stacked monomial algebras which are not selfinjective but ne…
We exploit singular equivalences between artin algebras, that are induced from certain functors between the stable module categories. Such functors are called pre-triangle equivalences. We construct two pre-triangle equivalences connecting the stable module category over a quadratic monomial algebra and the one over an algebra with radical square zero. Consequently, we obtain an explicit singular equivalence between the two algebras.
We describe the primitive central idempotents of the group algebra over a number field of finite monomial groups. We give also a description of the Wedderburn decomposition of the group algebra over a number field for finite strongly monomial groups. Further, for this class of group algebras, we describe when the number of simple components agrees with the number of simple components of the rational group algebra. Finally, we give a formula for the rank of the central units of the group ring over the ring of integers of a number field for a strongly monomial group.
UMP Monomial Algebras: Combinatorial and Homological Consequences
本文研究 UMP 单项式代数的分支图结构，证明其弱连通分支单侧连通，并建立二次单项式情形下 UMP 代数与特殊多重串代数的等价性，进而将 Gorenstein 投射模分类推广到单项式特殊多重串 UMP 代数。
Fomin-Zelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph and cluster complex are independent of the choice of coefficients. We confirm these conjectures for all skew-symmetric cluster algebras.
Linear independence of global monomials on positive spaces
本文证明了正空间上整体单项式的线性无关性，该方法基于洛朗展开的牛顿多面体研究，所得结果可应用于簇代数等相关代数，得到洛朗单项式性质及簇单项式线性无关性。
The stable category of Gorenstein-projective modules over a monomial algebra
本文研究任意单项式代数的分次 Gorenstein-投射模稳定范畴及其平移轨道范畴，证明二者分别与 Dynkin A 型路代数的有界导出范畴和自内射 Nakayama 代数的稳定模范畴三角等价。
We classify $n$-hereditary monomial algebras in three natural contexts: First, we give a classification of the $n$-hereditary truncated path algebras. We show that they are exactly the $n$-representation-finite Nakayama algebras classified by Vaso. Next, we classify partially the $n$-hereditary quadratic monomial algebras. In the case $n=2$, we prove that there are only two examples, provided that the preprojective algebra is a planar quiver with potential. The first one is a Nakayama algebra and the second one is obtained by mutating $\mathbb A_3\otimes_k \mathbb A_3$, where $\mathbb A_3$ is …
Our published paper contains an incorrect statement of a result due to Artin and Zhang. This corrigendum gives the correct statement of their result and includes a new result that allows us to use their result to prove our main theorem. Thus the main theorem of our published paper is correct as stated but its proof must be modified.
We compare the so-called clock condition to the gradability of certain differential modules over quadratic monomial algebras. For a stably hereditary algebra or a gentle one-cycle algebra, these considerations show that the orbit category of its bounded derived category with respect to a positive power of the shift functor is triangulated if and only if the algebra is piecewise hereditary.
On the monomial algebra associated to the monomial characters of a finite group
本文研究由有限群单项特征标生成的单项代数，指出其整闭包包含于关联Artin L-函数在特定点全纯的特征标生成的代数中，并讨论了超特征标理论情形。
A combinatorial characterisation of d-Koszul and (D,A)-stacked monomial algebras that satisfy (Fg)
本文给出了满足条件(Fg)的单项式d-Koszul代数及(D, A)-stacked代数的等价组合条件，并将d-Koszul代数的Yoneda代数描述推广至后者。
We describe an explicit crystal morphism between Nakajima monomials and monomials which give a realization of crystal bases for finite dimensional irreducible modules over the quantized enveloping algebra for Lie algebras of type A and C. This morphism provides a connection between arbitrary Nakajima monomials and Nakashima Kashiwara tableaux. This yields a translation of Nakajima monomials to the Littelmann path model. Furthermore, as an application of our results we describe an insertion scheme for Nakajima monomials compatible to the insertion scheme for tableaux.
G. E. Murphy showed in 1983 that the centre of every symmetric group algebra has an integral basis consisting of a specific set of monomial symmetric polynomials in the Jucys--Murphy elements. While we have shown in earlier work that the centre of the group algebra of S_3 has exactly three additional such bases, we show in this paper that the centre of the group algebra of S_4 has infinitely many bases consisting of monomial symmetric polynomials in Jucys--Murphy elements, which we characterize completely. The proof of this result involves establishing closed forms for coefficients of class su…
Kang, Kashiwara, Kim and Oh have proved that cluster monomials lie in the dual canonical basis, under a symmetric type assumption. This involves constructing a monoidal categorification of a quantum cluster algebra using representations of KLR algebras. We use a folding technique to generalise their results to all Lie types.
Centraliser algebras of monomial representations and applications in combinatorics
本文针对有限群单项式表示的中心化子代数，给出其基的显式构造并通过双陪集特征和构建特征表，将哈达玛矩阵相关理论纳入该框架，还应用格罗布纳基实现高对称复哈达玛矩阵的新分类。
Truncated shifted Yangians are a family of algebras which are natural quantizations of slices in the affine Grassmannian. We study the highest weight representations of these algebras. In particular, we conjecture that the possible highest weights for these algebras are described by product monomial crystals, certain natural subcrystals of Nakajima's monomials. We prove this conjecture in type A. We also place our results in the context of symplectic duality and prove a conjecture of Hikita in this situation.
In 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum $\mathfrak{gl}_n$ and its canonical basis. A key step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic construction of the canonical basis for modified quantum affine $\mathfrak{gl}_n$, which among other results used an earlier construction of monomial bases using Ringel-Hall algebra of the cyclic quiver. In this paper, we give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine $\mathfrak{gl}_n$.
Given a finite--dimensional monomial algebra $A$ we consider the trivial extension $TA$ and provide formulae, depending on the characteristic of the field, for the dimensions of the summands $HH_1(A)$ and $\Alt(DA)$ of the first Hochschild cohomology group $HH^1(TA)$. From these a formula for the dimension of $HH^1(TA)$ can be derived.
Monomial realizations and LS paths of fundamental representations for rank $2$ Kac-Moody algebras
本文针对秩 $2$ Kac-Moody 代数，显式构造了基本权的 Lakshmibai-Seshadri 路径集合与单项式实现之间的同构，并基于 Weyl 群给出了单项式实现的显式形式。
Cluster Monomials in Graph Laurent Phenomenon Algebras
本文研究图 Laurent 现象代数，证明其簇单项式构成线性基，并在底图为双向树时证明任意单项式的簇单项式展开系数非负。
Using combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, we give, for any monomial algebra $A$, an explicit description of its minimal model. This also provides us with formulas for a canonical $A_\infty$-structure on the Ext-algebra of the trivial $A$-module. We do this by exploiting the combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, and the algebraic discrete Morse theory of Jöllenbeck, Welker and Sköldberg. We then show how this result can be used to obtain models for algebras with a chosen Gröbner basis, and briefly outline …
We study a vertex operator algebra (VOA) V related to the M(3,p) Virasoro minimal series. This VOA reduces in the simplest case p=4 to the level two integrable vacuum module of $\hat{sl}_2$. On V there is an action of a commutative current a(z), which is an analog of the current e(z) of $\hat{sl}_2$. Our main concern is the subspace W generated by this action from the highest weight vector of V. Using the Fourier components of a(z), we present a monomial basis of W and a semi-infinite monomial basis of V. We also give a Gordon type formula for their characters.
We provide $\mathbb{N}$-filtrations on the negative part $U_q(\mathfrak{n}^-)$ of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}^-$. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an $\mathbb{N}$-filtration on any finite dimensional simple $\mathfrak{g}$-module. We prove for type $\tt{A}_n$, $\tt{C}_n…
The relationship between an algebra and its associated monomial algebra is investigated when at least one of the algebras is $d$-Koszul. It is shown that an algebra which has a reduced \grb basis that is composed of homogeneous elements of degree $d$ is $d$-Koszul if and only if its associated monomial algebra is $d$-Koszul. The class of 2-$d$-determined algebras and the class 2-$d$-Koszul algebras are introduced. In particular, it shown that 2-$d$-determined monomial algebras are 2-$d$-Koszul algebras and the structure of the ideal of relations of such an algebra is completely determined.
Singular equivalences induced by bimodules and quadratic monomial algebras
本文研究双模张量函子诱导奇异等价的条件，并构造显式双模以建立二次单项式代数与根平方零相关代数之间的带层级奇异等价。
Computing monomial bases in Lie theory using OSCAR
本文详细介绍如何使用计算机代数系统OSCAR计算单复李代数有限维模及偏旗簇齐次坐标环的单项式基，并随OSCAR能力进展持续更新。
We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac-Moody groups. We prove that all cluster monomials with g-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero-Chapoton description via quiver representations. In type A_1^{(1)}, we further show that elements of several canonical bases (generic, triangular, and…
The monomial expansion formula for Hall-Littlewood $P$-polynomials
本文通过Hecke代数方法推导了Hall-Littlewood多项式的Macdonald展开公式，并推广了Klostermann递归，为相关组合恒等式提供了新证明。
We prove that the basis of cluster monomials of a skew-symmetric cluster algebra A of finite type is the atomic basis of A. This means that an element of A is positive if and only if it has a non-negative expansion in the basis of cluster monomials. In particular cluster monomials are positive indecomposable, i.e. they cannot be written as a sum of positive elements.
Assume that $X= {x_1,...,x_g}$ is a finite alphabet and $K$ is a field. We study monomial algebras $A= K <X> /(W)$, where $W$ is an antichain of Lyndon words in $X$ of arbitrary cardinality. We find a Poincaré-Birkhoff-Witt type basis of $A$ in terms of its \emph{Lyndon atoms} $N$, but, in general, $N$ may be infinite. We prove that if $A$ has polynomial growth of degree $d$ then $A$ has global dimension $d$ and is standard finitely presented, with $d-1 \leq |W| \leq d(d-1)/2$. Furthermore, $A$ has polynomial growth iff the set of Lyndon atoms $N$ is finite. In this case $A$ has a $K$-basis $\…
In this paper, we study moduli spaces of representations of certain quivers with relations. For quivers without relations and other categories of homological dimension one, a lot of information is known about the cohomology of their moduli spaces of objects. On the other hand, categories of higher homological dimension remain more mysterious from this point of view, with few general methods. In this paper, we will see how some of the methods used to study quivers can be extended to work for representations of any (noncommutative) monomial algebra with relations of length two. In particular, we…
A unipotent triangular relationship is established between the dual standard monomial theoretic basis and canonical basis for the negative part of the quantized universal enveloping algebra of type A.
In this paper we define a two-variable, generic Hecke algebra, H, for each complex reflection group G(b,1,n). The algebra H specializes to the group algebra of G(b,1,n) and also to an endomorphism algebra of a representation of GL(n,q) induced from a solvable subgroup. We construct Kazhdan-Lusztig "R-polynomials" for H and show that they may be used to define a partial order on G(b,1,n). Using a generalization of Deodhar's notion of distinguished subexpressions we give a closed formula for the R-polynomials. After passing to a one-variable quotient of the ring of scalars, we construct Kazhdan-…
We mainly introduce an abstract pattern to study cluster algebras. Cluster algebras, generalized cluster algebras and Laurent phenomenon algebras are unified in the language of generalized Laurent phenomenon algebras (briefly, GLP algebras) from the perspective of Laurent phenomenon. In this general framework, we firstly prove that each positive and d-vector-positive GLP algebra has the proper Laurent monomial property and thus its cluster monomials are linearly independent. Skew-symmetric cluster algebras are verified to be d-vector-positive, which gives the affirmation of Conjecture [conjd] …
A conjecture on monomial realizations and polyhedral realizations for crystal bases
本文提出并部分证明了一个猜想，该猜想建立了可对称化 Kac-Moody 代数晶体基的多面体锥显式形式与单项式实现之间的联系，并在经典李代数、秩2 Kac-Moody 代数及经典仿射李代数情形下验证了其正确性。
We describe a recursive algorithm that produces an integral basis for the centre of the Hecke algebra of type A consisting of linear combinations of monomial symmetric polynomials of Jucys--Murphy elements. We also discuss the existence of integral bases for the centre of the Hecke algebra that consist solely of monomial symmetric polynomials of Jucys--Murphy elements. Finally, for n=3, we show that only one such basis exists for the centre of the Hecke algebra, by proving that there are exactly four bases for the centre of the corresponding symmetric group algebra which consist solely of mono…
Nakajima introduced the morphism of q,t-characters for finite dimensional representation of simply-laced quantum affine algebras : it is a t-deformation of the Frenkel-Reshetikhin's morphism of q-characters (sum of monomials in infinite variables). In math.QA/0212257 we generalized the construction of q,t-characters for non simply-laced quantum affine algebras. First in this paper we prove a conjecture of math.QA/0212257 : the monomials of q and q,t-characters of standard representations are the same in non simply-laced cases (the simply-laced cases were treated by Nakajima) and the coefficien…
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