Wreath Product Generalizations of the Triple $(S_{2n},H_{n},\phi)$ and Their Spherical Functions

Hiroshi Mizukawa

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Wreath Product Generalizations of the Triple $(S_{2n},H_{n},\phi)$ and Their Spherical Functions

Hiroshi Mizukawa

Representation Theory · Submitted 21 Aug 2009

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim It is shown that our triplet are always to be a Gelfand triple.

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arXiv ID0908.3056 (version 2)

Pith IDSDPTC2ZS, Pith's permanent record of this paper and its review

CitationHiroshi Mizukawa. "Wreath Product Generalizations of the Triple $(S_{2n},H_{n},\phi)$ and Their Spherical Functions." arXiv:0908.3056 (2009). https://pith.science/paper/SDPTC2ZS

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The symmetric group $S_{2n}$ and the hyperoctaheadral group $H_{n}$ is a Gelfand triple for an arbitrary linear representation $\phi$ of $H_{n}$. Their $\phi$-spherical functions can be caught as transition matrix between suitable symmetric functions and the power sums. We generalize this triplet in the term of wreath product. It is shown that our triplet are always to be a Gelfand triple. Furthermore we study the relation between their spherical functions and multi-partition version of the ring of symmetric functions.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The symmetric group $S_{2n}$ and the hyperoctaheadral group $H_{n}$ is a Gelfand triple for an arbitrary linear representation $\phi$ of $H_{n}$.	background 0.83	0.22	0.11	asserted 1.00
s1	Their $\phi$-spherical functions can be caught as transition matrix between suitable symmetric functions and the power sums.	result 0.68	0.49	0.09	asserted 0.96
s2	We generalize this triplet in the term of wreath product.	contribution 1.00	0.90	0.10	asserted 0.91
s3	It is shown that our triplet are always to be a Gelfand triple.	result 0.99	0.89	0.49	asserted 1.00
s4	Furthermore we study the relation between their spherical functions and multi-partition version of the ring of symmetric functions.	contribution 0.98	0.87	0.13	asserted 0.87

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Original abstract

The symmetric group $S_{2n}$ and the hyperoctaheadral group $H_{n}$ is a Gelfand triple for an arbitrary linear representation $\phi$ of $H_{n}$. Their $\phi$-spherical functions can be caught as transition matrix between suitable symmetric functions and the power sums. We generalize this triplet in the term of wreath product. It is shown that our triplet are always to be a Gelfand triple. Furthermore we study the relation between their spherical functions and multi-partition version of the ring of symmetric functions.

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3 arXiv papers cite this in 4 sentences

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To proceed with these sums, we will use a useful result by Howe [ref] (see also a result by Mizukawa, proposition 4.1 in [ref] , and also [ref] or a more recent use of it in [ref] ): ∑_γ∈S_n[S_2] χ^R(γ) = |S_n[S_2]| m^R where $m^R= 1$ if $R$ is an “even” partition, that is, all its row lengths are even, and $m^R=0$ otherwise.

On the counting of $O(N)$ tensor invariants · The counting as a Kronecker sum · 0.90
To tackle these sums, we use a result by Howe [ref] (see also [ref] ): $∑_γ∈S_n[S_2] χ^R(γ) = |S_n[S_2]| m^R$, where $m^R= 1$ if $R$ is an “even” partition, and $m^R=0$ otherwise.

On the counting tensor model observables as $U(N)$ and $O(N)$ classical invariants · Algebras and representation theoretic bases · 0.92

Cites it as support 1 paper

In fact, from existing group theory results we know that the irrep $[A]$ of $S_n[S_2]$ is always subduced once [ref] .

A basis for large operators in N=4 SYM with orthogonal gauge group · $SO(N)$ gauge theory · 0.97

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