Representation Theory of Finite Groups: Algebra and Arithmetic

Weintraub, Steven H.

doi:10.1090/gsm/059

Cited by 33 publications

(22 citation statements)

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“…It is clear that W (x, y, z) is the sign representation V (1,1,1,1) . Lastly, one easily checks that the composition factors of V (x, y) and V (2,2) ⊕ V (2,1,1) are the same; this follows, e.g., from an explicit computation using Brauer characters (see [20,Chpt. 7,Def.…”

Section: Examplesmentioning

confidence: 97%

On a notion of “Galois closure” for extensions of rings

Bhargava¹,

Satriano²

2014

J. Eur. Math. Soc.

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Section: Examplesmentioning

confidence: 97%

On a notion of “Galois closure” for extensions of rings

Bhargava¹,

Satriano²

2014

J. Eur. Math. Soc.

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“…The n-summand of h α # h β is, according to (9), in the right hand side of (9). We use Mackey's formula to interchange them (see [21]), as follows.…”

Section: The Heisenberg Product Of Complete Homogeneous Symmetric Funmentioning

confidence: 99%

The Heisenberg product: from Hopf algebras and species to symmetric functions

Aguiar

Santos

Moreira

2017

São Paulo J. Math. Sci.

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Abstract. Many related products and coproducts (e.g. Hadamard, Cauchy, Kronecker, induction, internal, external, Solomon, composition, Malvenuto-Reutenauer, convolution, etc.) have been defined in the following objects : species, representations of the symmetric groups, symmetric functions, endomorphisms of graded connected Hopf algebras, permutations, non-commutative symmetric functions, quasi-symmetric functions, etc. With the purpose of simplifying and unifying this diversity we introduce yet, another -non graded-product the Heisenberg product, that for the highest and lowest degrees produces the classical external and internal products (and their namesakes in different contexts). In order to define it, we start from the two opposite more general extremes: species in the "commutative context", and endomorphisms of Hopf algebras in the "non-commutative" environment. Both specialize to the space of commutative symmetric functions where the definitions coincide. We also deal with the different coproducts that these objects carry -to which we add the Heisenberg coproduct for quasi-symmetric functions-, and study their Hopf algebra compatibility particularly for symmetric and non commutative symmetric functions. We obtain combinatorial formulas for the structure constants of the new product that extend, generalize and unify results due to Garsia, Remmel, Reutenauer and Solomon. In the space of quasisymmetric functions, we describe explicitly the new operations in terms of alphabets.

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“…Briefly, the category A-Mod of A-modules is a 2-vector space with basis of absolutely simple objects any set of representatives of the irreducible modules. If A kÖG× the condition on k to be of characteristic zero or prime to the order of G is just the necessary and suficient condition for kÖG× to be semisimple (this is the famous Maschke's theorem; see Theorem 1.14 in Chapter 3 of [23]).…”

Section: Review On Projective Representations and Modules Over Arbitrmentioning

confidence: 99%

On the regular representation of an (essentially) finite 2-group

Elgueta

2011

Advances in Mathematics

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The regular representation of an essentially finite 2-group $\mathbb{G}$ in the 2-category $\mathbf{2Vect}_k$ of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all hom-categories in $\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})$ are 2-vector spaces under quite standard assumptions on the field $k$, and a formula giving the corresponding "intertwining numbers" is obtained which proves they are symmetric. Finally, it is shown that the forgetful 2-functor ${\boldmath$\omega$}:\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})\To\mathbf{2Vect}_k$ is representable with the regular representation as representing object. As a consequence we obtain a $k$-linear equivalence between the 2-vector space $\mathbf{Vect}_k^{\mathcal{G}}$ of functors from the underlying groupoid of $\mathbb{G}$ to $\mathbf{Vect}_k$, on the one hand, and the $k$-linear category $\mathcal{E} nd({\boldmath$\omega$})$ of pseudonatural endomorphisms of ${\boldmath$\omega$}$, on the other hand. We conclude that $\mathcal{E} nd({\boldmath$\omega$})$ is a 2-vector space, and we (partially) describe a basis of it.Postprint (published version

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Representation Theory of Finite Groups: Algebra and Arithmetic

Cited by 33 publications

References 0 publications

On a notion of “Galois closure” for extensions of rings

On a notion of “Galois closure” for extensions of rings

The Heisenberg product: from Hopf algebras and species to symmetric functions

On the regular representation of an (essentially) finite 2-group

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