Let V be a unitary vector space. Suppose H is a subgroup of the symmetric group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S_m$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>m</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> an irreducible unitary representation of H over a unitary space U. Denote by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$V_{\Lambda }(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>V</mi> <mi mathvariant="normal">Λ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the generalized symmetry class of tensors over V associated with H and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>. In this paper, we show that the vector space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$V_{\Lambda }(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>V</mi> <mi mathvariant="normal">Λ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has always an orthogonal basis consisting generalized decomposable symmetrized tensors. By determining the group module structure of generalized coset spaces, we give a group module structure for generalized orbital subspaces.

Orthogonal basis and group module structure for generalized symmetry classes of tensors

期刊论文
发表日期：

2025年10月1日

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Yousef Zamani,
Rasoul Khalilzadeh

摘要

Let V be a unitary vector space. Suppose H is a subgroup of the symmetric group $S_m$ $S_{m}$ and $\Lambda $ $Λ$ an irreducible unitary representation of H over a unitary space U. Denote by $V_{\Lambda }(H)$ $V_{Λ} (H)$ the generalized symmetry class of tensors over V associated with H and $\Lambda $ $Λ$ . In this paper, we show that the vector space $V_{\Lambda }(H)$ $V_{Λ} (H)$ has always an orthogonal basis consisting generalized decomposable symmetrized tensors. By determining the group module structure of generalized coset spaces, we give a group module structure for generalized orbital subspaces.