Exterior powers of representations of finite groups and integer-valued characters

Tomoyuki Tamura

Home/Archive/arXiv

arXiv · 1703.05497

Exterior powers of representations of finite groups and integer-valued characters

Tomoyuki Tamura

View original source ↗Read source document ↗Export metadata ↓

Abstract

For given representation of finite groups on a finite dimension complex vector space, we can define exterior powers of representations. In 1973, Knutson found one of methods of calculating the character of exterior powers of representations with properties of $λ$-rings. In this paper, we base this result of Knutson, and relate characters and elements of necklace rings, which were introduced by N.Metropolis and G-C.Rota in 1983, via a generating function of the character of exterior powers of representations. We focus on integer-valued characters and discuss a relation between integer-valued characters and element of necklace rings which has finite support and is contained in some images of truncated operations.

Explore related subjects

math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Tomoyuki Tamura. 2017-03-16. Exterior powers of representations of finite groups and integer-valued characters. https://arxiv.org/abs/1703.05497

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Minuscule Relations in Quantum $K$-Theory of Flag Varieties

We study the quantum $K$-theory of the flag variety $G/B$. For each minuscule fundamental weight $\varpi$, we construct an explicit relation in the torus-equivariant quantum $K$-theory $QK_T(G/B)$. The relation can be regarded as a quantum deformation of the character of the irreducible representation with highest weight $\varpi$.

math.RT↗

arXiv2026

A Gelfand model for the Okada algebra

In this paper, we construct a Gelfand model for the Okada algebra $O_n(X,Y)$ with generic parameters $X$ and $Y$, on the space of symmetric Okada arc diagrams using a conjugation-type action. The model is constructed inductively by identifying the Okada algebra as a diagram algebra and using the Jones basic construction to obtain a tower of algebras that are themselves Okada algebras at lower levels. We use the model to obtain all the irreducible representations of $O_n(X,Y)$, indexed by the elements of rank $n$ of the Young--Fibonacci lattice, and identify them with the cell modules of $O_n(X,Y)$.

math.RT↗

arXiv2026

Categorical Lie-Rinehart modules and Shen-Larsson functors

We develop a categorical framework for Lie-Rinehart monoids and their weak modules in a symmetric monoidal category. Using crossed homomorphisms, we construct a natural action of the monoidal category of modules over a Lie monoid on the category of weak Lie-Rinehart modules, thereby obtaining categorical versions of the Shen-Larsson functors. We further characterize the conditions under which the category of weak modules admits a monoidal structure and identify the corresponding condition for the associated functors to be strict monoidal. A dual theory for Lie- Rinehart comonoids and weak comodules is developed using cocrossed homomorphisms. Combining the module and comodule constructions, we obtain a bimodule category structure on the category of weak modules. Finally, we specialize the general framework to the symmetric monoidal category of super vector spaces, recovering Lie-Rinehart superalgebras and their associated Shen-Larsson-type constructions.

math.RT↗