We start with definitions of the general notions of the theory of $\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the spirit of approach of the papers \cite{VO,OV} to representation theory of symmetric groups. The main example is the classical - theory of the projective representations of symmetric groups.
Let $\Bbbk$ be a field, $G$ a group, and $(Q, I)$ a bound quiver. A map $W\colon Q_1 \to G$ is called a $G$-weight on $Q$, which defines a $G$-graded $\Bbbk$-category $\Bbbk(Q, W)$, and $W$ is called homogeneous if $I$ is a homogeneous ideal of the $G$-graded $\Bbbk$-category $\Bbbk(Q, W)$. Then we have a $G$-graded $\Bbbk$-category $\Bbbk(Q, I, W):= \Bbbk(Q, W)/I$. We can then form a smash product $\Bbbk(Q, I, W)\# G$ of $\Bbbk(Q, I, W)$ and $G$, which canonically defines a Galois covering $\Bbbk(Q, I, W)\# G \to \Bbbk(Q, I)$ with group $G$ (we will see that all such Galois coverings to $\Bbb…
We discuss finitely graded Iwanaga-Gorenstein (IG) algebras $A$ and representation theory of their (graded) Cohen-Macaulay (CM) modules. By quasi-Veronese algebra construction, in principle, we may reduce our study to the case where $A$ is a trivial extension algebra $A = Λ\oplus C$ with the grading $ \text{deg} Λ= 0, \ \text{deg} C = 1$. In the previous study, we gave a necessary and sufficient condition that $A$ is IG in terms of $Λ$ and $C$ by using derived tensor products and derived Homs. For simplicity, we assume that $Λ$ is of finite global dimension in the sequel. In this paper, we sho…
We give the graded Cartan determinants of the symmetric groups. Based on it, we propose a gradation of Hill's conjecture which is equivalent to Külshammer-Olsson-Robinson's conjecture on the generalized Cartan invariants of the symmetric groups.
Happel constructed a fully faithful functor $\mathcal{H} :\mathsf{D}^{\mathrm{b}}(\text{mod} \ Λ) \to \underline{\text{mod}}^{\Bbb{Z}} \ \text{T}(Λ)$ for a finite dimensional algebra $Λ$. He also showed that this functor $\mathcal{H}$ gives an equivalence precisely when $\text{gldim } Λ< \infty$. Thus if $\mathcal{H}$ gives an equivalence, then it provides a canonical tilting object $\mathcal{H} (Λ)$ of $\underline{\text{mod}}^{\mathbb{Z}} \ \text{T}(Λ)$. In this paper we generalize Happel's functor $\mathcal{H}$ in the case where $\text{T}(Λ)$ is replaced with a finitely graded IG algebra $A$…
Let $G$ be a complex, connected, reductive, algebraic group, and $χ:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $χ(\mathbb{C}^\times)$. In this paper, we study $G_0$-equivariant parity sheaves on $\mathfrak{g}_n$, under some assumptions on the field $\Bbbk$ and the group $G$. The assumption on $G$ holds for $GL_n$ and for any $G$, it recovers results of Lusztig in characteristic $0$. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspid…
We consider graded Cartan matrices of the symmetric groups and the Iwahori-Hecke algebras of type A, which have entries in the ring $\mathbb Z[v,v^{-1}]$. These matrices may also be interpreted as Gram matrices of the Shapovalov form on sums of weight spaces of a basic representation of an affine quantum group. We present a conjecture predicting the invariant factors of these matrices and give evidence for the conjecture by proving its implications under a localization and certain specializations of the ring $\mathbb Z[v,v^{-1}]$. This proves and generalizes a conjecture of Ando-Suzuki-Yamada …
On a derived equivalence classification of weakly symmetric algebras of Euclidean type
Shizuoka University（静冈大学）；Kyoto University（京都大学）；Osaka Central Advanced Mathematical Institute（大阪中央高等数学研究所）；Tokyo University of Science（东京理科大学）；Universidad de Murcia（穆尔西亚大学）
We use a representation of a graded twisted tensor product of $K[x]$ with $K[y]$ in $L(K^{\Bbb{N}_0})$ in order to obtain a nearly complete classification of these graded twisted tensor products via infinite matrices. There is one particular example and three main cases: quadratic algebras classified by Conner and Goetz, a family called $A(n,d,a)$ with the $n+1$-extension property for $n\ge 2$, and a third case, not fully classified, which contains a family $B(a,L)$ parameterized by quasi-balanced sequences.
Let $A=\oplus_{i\in \mathbb{Z}}A_i$ be a finite dimensional graded symmetric cellular algebra with a homogeneous symmetrizing trace of degree $d$. We prove that $A_{-d}$ contains the Higman ideal $H(A)$ of the center of $A$ and $\dim H(A)\leq \dim A_{0}$ if $d\neq 0$, and provide a semisimplicity criterion of $A$ in terms of the centralizer of $A_0$, which is a graded version of \cite[Theorem 3.2]{L}.
We review some recent advances in modular representation theory of symmetric groups and related Hecke algebras. We discuss connections with Khovanov-Lauda-Rouquier algebras and gradings on the blocks of the group algebras $FΣ_n$, which these connections reveal; graded categorification and connections with quantum groups and crystal bases; modular branching rules and the Mullineaux map; graded cellular structure and graded Specht modules; cuspidal systems for affine KLR algebras and imaginary Schur-Weyl duality, which connects representation theory of these algebras to the usual Schur algebras …
We classify all graded compatible left-symmetric algebraic structures on high rank Witt algebras, and classify all non-graded ones satisfying a minor condition. Furthermore, graded compatible left-symmetric algebraic structures on high rank Virasoro algebras are also classified.
We introduce the notion of stable representations, -- it is a new class of the representations of a certain class of groups which defined with positive definite functions which generalize the classical notion of the characters (or trace). We give the complete description of this class for infinite symmetric group ${\frak S}_{\Bbb N}$. It happened the family of the stable representations of ${\frak S}_{\Bbb N}$ coincides with the set of representations of the components (left or right) of the admissible representations of the double-group in the sense of Ol'shanski. In particular all these are …
Dave Benson conjectured in 2020 that if $G$ is a finite $2$-group and $V$ is an odd-dimensional indecomposable representation of $G$ over an algebraically closed field $\Bbbk$ of characteristic $2$, then the only odd-dimensional indecomposable summand of $V \otimes V^*$ is the trivial representation $\Bbbk$. This would imply that a tensor power of an odd-dimensional indecomposable representation of $G$ over $\Bbbk$ has a unique odd-dimensional summand. Benson has further conjectured that, given such a representation $V$, the function sending a positive integer $n$ to the dimension of the uniqu…
Brundan and Kleshchev introduced graded decomposition numbers for representations of cyclotomic Hecke algebras of type $A$, which include group algebras of symmetric groups. Graded decomposition numbers are certain Laurent polynomials, whose values at 1 are the usual decomposition numbers. We show that in quantum characteristic 2 every such polynomial has non-zero coefficients either only in odd or only in even degrees. As a consequence, we find the first examples of graded decomposition numbers of symmetric groups with non-zero coefficients in some negative degrees.
We give uniform formulas for the branching rules of level 1 modules over orthogonal affine Lie algebras for all conformal pairs associated to symmetric spaces. We also provide a combinatorial intepretation of these formulas in terms of certain abelian subalgebras of simple Lie algebras.
The main result of this paper shows that, over large enough fields of characteristic different from $2$, the alternating Hecke algebras are $\mathbb{Z}$-graded algebras that are isomorphic to fixed-point subalgebras of the quiver Hecke algebra of the symmetric group $\mathfrak{S}_n$. As a special case, this shows that the group algebra of the alternating group, over large enough fields of characteristic different from $2$, is a $\mathbb{Z}$-graded algebra. We give a homogeneous presentation for these algebras, compute their graded dimension and show that the blocks of the quiver Hecke algebras…
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $\ell$ over an algebraically closed field $\Bbbk$ of characteristic zero, and let $(e,h,f)$ be an $\mathfrak{sl}_2$-triple of g. Denote by $\mathfrak{g}^{e}$ the centralizer of $e$ in $\mathfrak{g}$ and by ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ the algebra of symmetric invariants of $\mathfrak{g}^{e}$. We say that $e$ is good if the nullvariety of some $\ell$ homogenous elements of ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ in $(\mathfrak{g}^{e})^{*}$ has codimension $\ell$. If $e$ is good then ${\rm S}(\mathfrak…
We consider the totally nonfree (TNF) action of a groups and the corresponding adjoint invariant (AD) measures on the lattices of the subgroups of the given group. The main result is the description of all adjoint-invariant and TNF-measures on the lattice of subgroups of the infinite symmetric group $S_{\Bbb N}$. The problem is closely related to the theory of characters and factor representations of groups.
Recently, the first two authors have defined a Z-grading on group algebras of symmetric groups and more generally on the cyclotomic Hecke algebras of type G(l,1,d). In this paper we explain how to grade Specht modules over these algebras.
We use graded Specht modules to calculate the graded decomposition numbers for the Iwahori-Hecke algebra of the symmetric group over a field of characteristic zero at a root of unity. The algorithm arrived at is the Lascoux-Leclerc-Thibon algorithm in disguise. Thus we interpret the algorithm in terms of graded representation theory.
We consider the universal pivotal, symmetric, monoidal, $\Bbbk$-linear category, generated by a Schurian object with a skew-symmetric multiplication, and study some of its quotients. We show that these quotients give rise to either vector product algebras or representation categories of exceptional Lie algebras.
We consider symmetric pairs of Lie superalgebras which are strongly reductive and of even type, and introduce a graded Harish-Chandra homomorphism. We prove that its image is a certain explicit filtered subalgebra of the Weyl invariants on a Cartan subspace whose associated graded is the image of Chevalley's restriction map on symmetric invariants. This generalises results of Harish-Chandra and V. Kac, M. Gorelik.
In this work we study the classes of epsilon and nearly epsilon-strongly graded rings by a group $G$. In particular, we extend Dade's theorem to the realm of nearly epsilon-strongly graded rings. Moreover, we introduce the category SIM$S-$gr of symmetrically graded modules and use it to present a new characterization of strongly graded rings. A functorial approach is used to obtain a characterization of epsilon-strongly graded rings. Finally, we determine conditions for which an epsilon-strongly graded ring can be written as a direct sum of strongly graded rings and a trivially graded ring.
Let ${\goth g}$ be a finite dimensional complex reductive Lie algebra and $\dv ..$ an invariant non degenerated bilinear form on ${\goth g}\times {\goth g}$ which extends the Killing form of $[{\goth g},{\goth g}]$. We define the homology complex $C_{\bullet}({\goth g})$. Its space is the algebra $\tk {\Bbb C}{\e Sg}\tk {\Bbb C}{\e Sg}\ex {}{\goth g}$ where $\e Sg$ and $\ex {}{\goth g}$ are the symmetric and exterior algebras of ${\goth g}$. The differential of $C_{\bullet}({\goth g})$ is the $\tk {\Bbb C}{\e Sg}\e Sg$-derivation which associates to the element $v$ of ${\goth g}$ the function …
An $r$-dual tower of groups is a nested sequence of finite groups, like the symmetric groups, whose Bratteli diagram forms an $r$-dual graded graph. Miller and Reiner introduced a special case of these towers in order to study the Smith forms of the up and down maps in a differential poset. Agarwal and the author have also used these towers to compute critical groups of representations of groups appearing in the tower. In this paper I prove that when $r$ is one or prime, wreath products of a fixed group with the symmetric groups are the only $r$-dual tower of groups, and conjecture that this i…
We show that for any finite $p$-group $P$ of rank at least 2 and any algebraically closed field $k$ of characteristic $p$ the graded center $Z^*(\modbar(kP))$ of the stable module category of finite-dimensional $kP$-modules has infinite dimension in each odd degree, and if $p=2$ also in each even degree. In particular, this provides examples of symmetric algebras $A$ for which $Z^0(\modbar(A))$ is not finite-dimensional, answering a question raised in [10]
We introduce a new set of $q$-difference operators acting as raising operators on a family of symmetric polynomials which are characters of graded tensor products of current algebra ${\mathfrak g}[u]$ KR-modules \cite{FL} for ${\mathfrak g}=A_r$. These operators are generalizations of the Kirillov-Noumi \cite{kinoum} Macdonald raising operators, in the dual $q$-Whittaker limit $t\to\infty$. They form a representation of the quantum $Q$-system of type $A$ \cite{qKR}. This system is a subalgebra of a quantum cluster algebra, and is also a discrete integrable system whose conserved quantities, an…
We introduce and study potentials, mutations and Jacobian algebras in the framework of tensor algebras associated with symmetrizable dualizing pairs of bimodules on a symmetric algebra over any commutative ground ring. The graded context is also considered by starting from graded bimodules, and the classical non simply-laced context of modulated quivers with potentials is a particular case. The study of potentials in this framework is related to symmetrically separable algebras, and we have two kinds of potentials: the symmetric and the non symmetric ones. When the Casimir ideal of the symmetr…
We construct a non-commutative, non-cocommutative, graded bialgebra $\mathbfΠ$ with a basis indexed by the permutations in all finite symmetric groups. Unlike the formally similar Malvenuto-Poirier-Reutenauer Hopf algebra, this bialgebra does not have finite graded dimension. After giving formulas for the product and coproduct, we show that there is a natural morphism from $\mathbfΠ$ to the algebra of quasi-symmetric functions, under which the image of a permutation is its associated Stanley symmetric function. As an application, we use this morphism to derive some new enumerative identities. …
