Induced operators on the generalized symmetry classes of tensors
محل انتشار: فصلنامه تئوری گروهی، دوره: 10، شماره: 4
سال انتشار: 1400
نوع سند: مقاله ژورنالی
زبان: انگلیسی
مشاهده: 265
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شناسه ملی سند علمی:
JR_THEGR-10-4_005
تاریخ نمایه سازی: 14 اردیبهشت 1400
چکیده مقاله:
Let $V$ be a unitary space. Suppose $G$ is a subgroup of the symmetric group of degree $m$ and $\Lambda$ is an irreducible unitary representation of $G$ over a vector space $U$. Consider the generalized symmetrizer on the tensor space $U\otimes V^{\otimes m}$, $ S_{\Lambda}(u\otimes v^{\otimes})=\dfrac{۱}{|G|}\sum_{\sigma\in G}\Lambda(\sigma)u\otimes v_{\sigma^{-۱}(۱)}\otimes\cdots\otimes v_{\sigma^{-۱}(m)} $ defined by $G$ and $\Lambda$. The image of $U\otimes V^{\otimes m}$ under the map $S_\Lambda$ is called the generalized symmetry class of tensors associated with $G$ and $\Lambda$ and is denoted by $V_\Lambda(G)$. The elements in $V_\Lambda(G)$ of the form $S_{\Lambda}(u\otimes v^{\otimes})$ are called generalized decomposable tensors and are denoted by $u\circledast v^{\circledast}$. For any linear operator $T$ acting on $V$, there is a unique induced operator $K_{\Lambda}(T)$ acting on $V_{\Lambda}(G)$ satisfying $ K_{\Lambda}(T)(u\otimes v^{\otimes})=u\circledast Tv_{۱}\circledast \cdots \circledast Tv_{m}. $ If $\dim U=۱$, then $K_{\Lambda}(T)$ reduces to $K_{\lambda}(T)$, induced operator on symmetry class of tensors $V_{\lambda}(G)$. In this paper, the basic properties of the induced operator $K_{\Lambda}(T)$ are studied. Also some well-known results on the classical Schur functions will be extended to the case of generalized Schur functions.
کلیدواژه ها:
irreducible representation ، generalized Schur function ، generalized symmetrizer ، generalized symmetry class of tensors ، induced operator
نویسندگان
Gholamreza Rafatneshan
Department of Mathematics, Faculty of Basic Sciences, Sahand University of Technology, P.O. Box ۵۱۳۳۵/۱۹۹۶, Tabriz, Iran
Yousef Zamani
Department of Mathematics, Faculty of Basic Sciences, Sahand University of Technology, P.O. Box ۵۱۳۳۵/۱۹۹۶, Tabriz, Iran