In the first part of this paper we present explicit formulas for primitive idempotents in arbitrary Frobenius algebras using the entries of representing matrices coming from projective indecomposable modules with respect to a certain choice of basis. The proofs use a generalisation of the well known Frobenius-Schur relations for semisimple algebras.
The second part of this paper considers $\Oh$-free $\Oh$-algebras of finite $\Oh$-rank over a discrete valuation ring $\Oh$ and their decomposition maps under modular reduction modulo the maximal ideal of $\Oh$, thereby studying the modular representation theory of such algebras.
Using the formulas from the first part we derive general criteria for such a decomposition map to be an isomorphism that preserves the classes of simple modules involving explicitly known matrix representations on projective indecomposable modules.
Finally we show how this approach could eventually be used to attack a conjecture by Gordon James in the formulation of Meinolf Geck for Iwahori-Hecke-Algebras, provided the necessary matrix representations on projective indecomposable modules could be constructed explicitly.
We give a description of the primitive central idempotents of the rational group algebra QG of a finite group G. Such a description is already investigated by Jespers, Olteanu and del Río, but some unknown scalars are involved. Our description also gives answers to their questions.
We introduce and study the notion of pseudo-Frobenius graded algebra with enough idempotents, showing that it follows the pattern of the classical concept of pseudo-Frobenius (PF) and Quasi-Frobenius (QF) rings, in particular finite dimensional self-injective algebras, as studied by Nakayama, Morita, Faith, Tachikawa, etc. We show that such an algebra is characterized by the existence of a graded Nakayama form. Moreover, we prove that the pseudo-Frobenius property is preserved and reflected by covering functors, a fact that makes the concept useful in Representation Theory.
An approach to representations of finite groups is presented without recourse to character theory. Considering the group algebra C[G] as an algebra of linear maps on C[G] (by left multiplication), we derive the primitive central idempotents as a simultaneous eigenbasis of the centre, Z(C[G]). We apply this framework to obtain the irreducible representations of a class of finite meta-abelian groups. In particular, we give a general construction of the isomorphism between simple blocks of C[G] and the corresponding matrix algebra where G can be any finite group.
In this paper we give an explicit description of primitive central idempotents of rational group algebras of finite abelian groups using long presentation, and determine their Wedderburn decompositions.
We construct a recursive formula for a complete system of primitive orthogonal idempotents for any $R$-trivial monoid. This uses the newly proved equivalence between the notions of $R$-trivial monoid and weakly ordered monoid.
We provide an explicit construction for a complete set of orthogonal primitive idempotents of finite group algebras over nilpotent groups. Furthermore, we give a complete set of matrix units in each simple epimorphic image of a finite group algebra of a nilpotent group.
Formulas for the primitive idempotents of the trivial source algebra, in characteristic zero, have been given by Boltje and Bouc--Thévenaz. We shall give another formula for those idempotents, expressing them as linear combinations of the elements of a canonical basis for the integral ring. The formula is an inversion formula analogous to the Gluck--Yoshida formula for the primitive idempotents of the Burnside algebra. It involves all the irreducible characters of all the normalizers of $p$-subgroups. As a corollary, we shall show that the linearization map from the monomial Burnside ring has …
Let $G$ be a finite solvable group. Then $G$ always has a useful presentation, which we call a "long presentation". Using a "long presentation" of $G$, we present an inductive method of constructing the irreducible representations of $G$ over $\mathbb{C}$ and computing the primitive central idempotents of the complex group algebra $\mathbb{C}[G]$. For a finite abelian group, we present a systematic method of constructing the irreducible representations over a field of characteristic either $0$ or prime to order of the group and also a systematic method of computing the primitive central idempo…
We determine the blocks, i.e., the primitive central idempotents, of the bifree double Burnside ring $B^Δ(G,G)$ and the left-free double Burnside ring $B^{\trl}(G,G)$, as well as the primitive central idempotents of the algebras arising from scalar extension to $\QQ$.
Let $Γ$ denote a distance-regular graph with diameter $D\geq 3$ and Bose-Mesner algebra $M$. For $θ\in C\cup \infty$ we define a 1 dimensional subspace of $M$ which we call $M(θ)$. If $θ\in C$ then $M(θ)$ consists of those $Y$ in $M$ such that $(A-θI)Y\in C A_D$, where $A$ (resp. $A_D$) is the adjacency matrix (resp. $D$th distance matrix) of $Γ.$ If $θ= \infty$ then $M(θ)= C A_D$. By a {\it pseudo primitive idempotent} for $θ$ we mean a nonzero element of $M(θ)$. We use pseudo primitive idempotents to describe the irreducible modules for the Terwilliger algebra, that are thin with endpoint on…
The authors of [Primitive orthogonal idempotents for R-trivial monoids, Journal of Algebra] provide an algorithm for finding a complete system of primitive orthogonal idempotents for CM, where M is any finite R-trivial monoid. Their method relies on a technical result stating that R-trivial monoids are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an R-trivial monoid may be realised by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where L-tr…
We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An immediate consequence is a well-known result of Roquette on the Schur indices of the simple components of group algebras of finite nilpotent groups. As an application, we obtain that the unit group of the integral group ring $\Z G$ of a finite nilpotent group $G$ has a subgroup of finite index that is generated by three nilpotent groups for which we have an ex…
We provide a method for constructing central idempotents in the Brauer algebra relating to the splitting of certain short exact sequences. We also determine some of the primitive central idempotents, and relate properties of the idempotents to known facts about the representation theory of the algebra.
Given a morphism of (small) groupoids with injective object map, we provide sufficient and necessary conditions under which the induction and co-induction functors between the categories of linear representations are naturally isomorphic. A morphism with this property is termed a Frobenius morphism of groupoids. As a consequence, an extension by a subgroupoid is Frobenius if and only if each fibre of the (left or right) pull-back biset has finitely many orbits. Our results extend and clarify the classical Frobenius reciprocity formulae in the theory of finite groups, and characterize Frobenius…
The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots p_{n}$, where $p_{i}$'s are primes, there always exists a subnormal series: $\langle {e} \rangle = G_{o} < G_{1} < \dots < G_{n} = G$ such that $G_{i}/G_{i-1}$ is isomorphic to a cyclic group of order $p_{i}$, $i = 1,2,\dots,n$. Associated with this series, there exists a system of generators consisting $n$ elements $x_{1}, x_{2}, \dots, x_{n}$ (say), such that $G_{i} = \langle x_{1}, x_{2}…
Any multiplicity-free family of finite dimensional algebras has a canonical complete set of of pairwise orthogonal primitive idempotents in each level. We give various methods to compute these idempotents. In the case of symmetric group algebras over a field of characteristic zero, the set of canonical idempotents is precisely the set of seminormal idempotents constructed by Young. As an example, we calculate the canonical idempotents for semisimple Brauer algebras.
We give a new construction of primitive idempotents of the Hecke algebras associated with the symmetric groups. The idempotents are found as evaluated products of certain rational functions thus providing a new version of the fusion procedure for the Hecke algebras. We show that the normalization factors which occur in the procedure are related to the Ocneanu--Markov trace of the idempotents.
We show that the quantum Frobenius morphism constructed by Lusztig in the setting of the quantum enveloping algebra specialized at a root of unity admits a multiplicative splitting (non unital). We also find a basis of the toral part of the small quantum algebra consisting of pairwise orthogonal idempotents summing up to 1, and likewise in the modular case of the algebra of distributions for a semisimple algebraic group.
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.
We describe what it means for an algebra to be internally d-Calabi-Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d-1)-cluster-tilting objects in certain stably (d-1)-Calabi-Yau Frobenius categories, as observed by Keller-Reiten. We show that an internally d-Calabi-Yau algebra satisfying mild additional assumptions can be realised as the endomorphism algebra of a (d-1)-cluster-tilting object in a Frobenius category. Moreover, if the algebra satisfies a stronger 'bimodule' internally d-Calabi-Yau condition, this Frobenius category is stably …
In this note, we prove that if $Λ$ is an Artin algebra with a simple module $S$ of finite projective dimension, then the finiteness of the finitistic dimension of $Λ$ implies that of $(1-e)Λ(1-e)$ where $e$ is the primitive idempotent supporting $S$. We derive some consequences of this. In particular, we recover a result of Green-Solberg-Psaroudakis: if $Λ$ is the quotient of a path algebra by an admissible ideal $I$ whose defining relations do not involve a certain arrow $α$, then the finitistic dimension of $Λ$ is finite if and only if the finitistic dimension of $Λ/ΛαΛ$ is finite.
The primitive idempotents of the generic degenerate cycloctomic Hecke algebras are derived by consecutive evaluations of a certain rational function. This rational function depends only on the Specht modules and the normalization factors are the weights of the Brundan-Kleshchev trace.
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.
For a cellular algebra $\A$ with a cellular basis $\ZC$, we consider a decomposition of the unit element $1_\A$ into orthogonal idempotents (not necessary primitive) satisfying some conditions. By using this decomposition, the cellular basis $\ZC$ can be partitioned into some pieces with good properties. Then by using a certain map $\a$, we give a coarse partition of $\ZC$ whose refinement is the original partition. We construct a Levi type subalgebra $\aA$ of $\A$ and its quotient algebra $\oA$, and also construct a parabolic type subalgebra $\tA$ of $\A$, which contains $\aA$ with respect to…
Motivated by understanding the Broué's abelian defect group conjecture from algebraic point of view, we consider the question of how to lift a stable equivalence of Morita type between arbitrary finite dimensional algebras to a derived equivalence. In this paper, we present a machinery to solve this question for a class of stable equivalences of Morita type. In particular, we show that every stable equivalence of Morita type between Frobenius-finite algebras over an algebraically closed field can be lifted to a derived equivalence. %Thus Frobenius-finite algebras share many common %invariants …
We classify the primitive idempotents of the $p$-local complex representation ring of a finite group $G$ in terms of the cyclic subgroups of order prime to $p$ and show that they all come from idempotents of the Burnside ring. Our results hold without adjoining roots of unity or inverting the order of $G$, thus extending classical structure theorems. We then derive explicit group-theoretic obstructions for tensor induction to be compatible with the resulting idempotent splitting of the representation ring Mackey functor. Our main motivation is an application in homotopy theory: we conclude tha…
Let $k$ be an algebraically closed field and $A$ be a (left and right) Noetherian associative $k$-algebra. Assume further that $A$ is either positively graded or semiperfect (this includes the class of finite dimensional $k$-algebras, and $k$-algebras that are finitely generated modules over a Noetherian central Henselian ring). Let $e$ be a primitive idempotent of $A$, which we assume is of degree $0$ if $A$ is positively graded. We consider the idempotent subalgebra $Γ= (1-e)A(1-e)$ and $S_e$ the simple right $A$-module $S_e = eA/e{\rm rad}A$, where ${\rm rad}A$ is the Jacobson radical of $A…
We describe an algorithm for computing the complete set of primitive orthogonal idempotents in the centralizer ring of the permutation representation of a wreath product. This set of idempotents determines the decomposition of the representation into irreducible components. In the formalism of quantum mechanics, these idempotents are projection operators into irreducible invariant subspaces of the Hilbert space of a multipartite quantum system. The C implementation of the algorithm constructs irreducible decompositions of high-dimensional representations of wreath products. Examples of computa…
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.
Let $\mathcal H$ be the class of algebras verifying Han's conjecture. In this paper we analyse two types of algebras with the aim of providing an inductive step towards the proof of this conjecture. Firstly we show that if an algebra $Λ$ is triangular with respect to a system of non necessarily primitive idempotents, and if the algebras at the idempotents belong to $\mathcal H$, then $Λ$ is in $\mathcal H$. Secondly we consider a $2\times 2$ matrix algebra, with two algebras on the diagonal, two projective bimodules in the corners, and zero corner products. They are not triangular with respect…
In this note, we first prove that the degenerate cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}$ is cellular by constructing an explicit cellular basis. We then develop the fusion procedure for $Y_{r,n}^{d}$, that is, a complete set of pairwise orthogonal primitive idempotents for $Y_{r,n}^{d}$ is defined by consecutive evaluations of a certain rational function.
A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element.
We show that all primitive idempotents for the Brauer algebra B_n(w) can be found by evaluating a rational function in several variables which has the form of a product of R-matrix type factors. This provides an analogue of the fusion procedure for B_n(w).
The classical characteristic map associates symmetric functions to characters of the symmetric groups. There are two natural analogues of this map involving the Brauer algebra. The first of them relies on the action of the orthogonal or symplectic group on a space of tensors, while the second is provided by the action of this group on the symmetric algebra of the corresponding Lie algebra. We consider the second characteristic map both in the orthogonal and symplectic case, and calculate the images of central idempotents of the Brauer algebra in terms of the Schur polynomials. The calculation …
A complete system of primitive pairwise orthogonal idempotents for the Coxeter groups of type $B$ and, more generally, for the complex reflection groups $G(m,1,n)$ is constructed by a sequence of evaluations of a rational function in several variables with values in the group ring. The evaluations correspond to the eigenvalues of the two arrays of Jucys--Murphy elements.
We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux: fixing two integers n>2 and k>0 we shift an ordinary skew diagram of two partitions, viewed as a subset of the two-dimensional integer lattice, by the period vector (n,-k). Imposing a periodicity condition one defines cylindric skew tableaux as a map from the periodically continued skew diagram into the integers. The resulting cylindric Macdonald functions appear in the coproduct of a commutative Frobenius algebra,…
