10 research outputs found
Words and polynomial invariants of finite groups in non-commutative variables
Let V be a complex vector space with basis {x_1,x_2,...,x_n} and G be a
finite subgroup of GL(V). The tensor algebra T(V) over the complex is
isomorphic to the polynomials in the non-commutative variables x_1, x_2,...,
x_n with complex coefficients. We want to give a combinatorial interpretation
for the decomposition of T(V) into simple G-modules. In particular, we want to
study the graded space of invariants in T(V) with respect to the action of G.
We give a general method for decomposing the space T(V) into simple modules in
terms of words in a Cayley graph of the group G. To apply the method to a
particular group, we require a homomorphism from a subalgebra of the group
algebra into the character algebra. In the case of G as the symmetric group, we
give an example of this homomorphism from the descent algebra. When G is the
dihedral group, we have a realization of the character algebra as a subalgebra
of the group algebra. In those two cases, we have an interpretation for the
graded dimensions of the invariant space in term of those words
Invariants in Non-Commutative Variables of the Symmetric and Hyperoctahedral Groups
We consider the graded Hopf algebra of symmetric functions with non-commutative variables, which is analogous to the algebra of the ordinary symmetric functions in commutative variables. We give formulaes for the product and coproduct on some of the analogues of the bases and expressions for a shuffle product on . We also consider the invariants of the hyperoctahedral group in the non-commutative case and a state a few results
Words and polynomial invariants of finite groups in non-commutative variables
Let be a complex vector space with basis and be a finite subgroup of . The tensor algebra over the complex is isomorphic to the polynomials in the non-commutative variables with complex coefficients. We want to give a combinatorial interpretation for the decomposition of into simple -modules. In particular, we want to study the graded space of invariants in with respect to the action of . We give a general method for decomposing the space into simple -module in terms of words in a particular Cayley graph of . To apply the method to a particular group, we require a surjective homomorphism from a subalgebra of the group algebra into the character algebra. In the case of the symmetric group, we give an example of this homomorphism from the descent algebra. When is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions of the invariant space in term of those words
Words and Noncommutative Invariants of the Hyperoctahedral Group
Let be the hyperoctahedral group acting on a complex vector space . We present a combinatorial method to decompose the tensor algebra on into simple modules via certain words in a particular Cayley graph of . We then give combinatorial interpretations for the graded dimension and the number of free generators of the subalgebra of invariants of , in terms of these words, and make explicit the case of the signed permutation module. To this end, we require a morphism from the Mantaci-Reutenauer algebra onto the algebra of characters due to Bonnafé and Hohlweg.Soit le groupe hyperoctaédral agissant sur un espace vectoriel complexe . Nous présentons une méthode combinatoire donnant la décomposition de l'algèbre des tenseurs sur en modules simples via certains mots dans un graphe de Cayley donné. Nous donnons ensuite des interprétations combinatoires pour la dimension graduée et le nombre de générateurs libres de la sous-algèbre des invariants de , en termes de ces mots, et explicitons le cas du module de permutation signé. À cette fin, nous utilisons un morphisme entre l'algèbre de Mantaci-Reutenauer et l'algèbre des caractères introduit par Bonnafé et Hohlweg
Invariants in Non-Commutative Variables of the Symmetric and Hyperoctahedral Groups
We consider the graded Hopf algebra of symmetric functions with non-commutative variables, which is analogous to the algebra of the ordinary symmetric functions in commutative variables. We give formulaes for the product and coproduct on some of the analogues of the bases and expressions for a shuffle product on . We also consider the invariants of the hyperoctahedral group in the non-commutative case and a state a few results.Nous considérons l'algèbre de Hopf graduée des fonctions symétriques en variables non-commutatives, qui est analogue à l'algèbre des fonctions symétriques en variables commutatives. Nous donnons des formules pour le produit et coproduit sur certaines des bases analogues à celles de , ainsi qu'une expression pour le produit sur . Nous considérons aussi les invariants du groupe hyperoctaédral dans le cas non-commutatif et énonçons quelques résultats
Words and polynomial invariants of finite groups in non-commutative variables
Let be a complex vector space with basis and be a finite subgroup of . The tensor algebra over the complex is isomorphic to the polynomials in the non-commutative variables with complex coefficients. We want to give a combinatorial interpretation for the decomposition of into simple -modules. In particular, we want to study the graded space of invariants in with respect to the action of . We give a general method for decomposing the space into simple -module in terms of words in a particular Cayley graph of . To apply the method to a particular group, we require a surjective homomorphism from a subalgebra of the group algebra into the character algebra. In the case of the symmetric group, we give an example of this homomorphism from the descent algebra. When is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions of the invariant space in term of those words.Soit V un espace vectoriel complexe de base et un sous-groupe fini de . L'algèbre des tenseurs de sur les complexes est isomorphe aux polynômes à coefficients complexes en variables non-commutatives . Nous voulons donner une décomposition de en -modules simples de manière combinatoire. Plus particulièrement, nous étudions l'espace gradué des invariants de sous l'action de . Nous présentons une méthode générale donnant la décomposition de en modules simples via certains mots dans un graphe de Cayley donné. Pour appliquer la méthode à un groupe particulier, nous avons besoin d'un homomorphisme surjectif entre une sous-algèbre de l'algèbre de groupe et l'algèbre des caractères. Pour le cas du groupe symétrique, nous donnons un exemple de cet homomorphisme qui provient de la théorie de l'algèbre des descentes. Pour le groupe diédral, nous avons une réalisation de l'algèbre des caractères comme une sous-algèbre de l'algèbre de groupe. Dans ces deux cas, nous avons une interprétation des dimensions graduées de l'espace des invariants en terme de ces mots