Linear transformations that are tridiagonal with respect to the three decompositions for an LR triple

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• 作者：Kazumasa Nomura ^{knomura@pop11.odn.ne.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：17B37 ; 15A21
• 刊名：Linear Algebra and its Applications
• 出版年：2015
• 出版时间：1 December 2015
• 年：2015
• 卷：486
• 期：Complete
• 页码：173-203
• 全文大小：506 K

文摘

Recently, Paul Terwilliger introduced the notion of a lowering–raising (or LR) triple, and classified the LR triples. An LR triple is defined as follows. Fix an integer mmlsi1" class="mathmlsrc">d≥0$d\ge 0$, a field mmlsi177" class="mathmlsrc">F$\mathbb{F}$, and a vector space V   over mmlsi177" class="mathmlsrc">F$\mathbb{F}$ with dimension mmlsi3" class="mathmlsrc">d+1$d+1$. By a decomposition of V   we mean a sequence mmlsi178" class="mathmlsrc">${\{V_i\}}_{i=0}^d$ of 1-dimensional subspaces of V whose sum is V. For a linear transformation A from V to V, we say A   lowers mmlsi178" class="mathmlsrc">${\{V_i\}}_{i=0}^d$ whenever mmlsi5" class="mathmlsrc">AV_i=V_i−1$A{V}_i=V_{i-1}$ for mmlsi179" class="mathmlsrc">0≤i≤d$0\le i\le d$, where mmlsi7" class="mathmlsrc">V₋₁=0$V_{-1}=0$. We say A   raises mmlsi178" class="mathmlsrc">${\{V_i\}}_{i=0}^d$ whenever mmlsi8" class="mathmlsrc">AV_i=V_i+1$A{V}_i=V_{i+1}$ for mmlsi179" class="mathmlsrc">0≤i≤d$0\le i\le d$, where mmlsi9" class="mathmlsrc">V_d+1=0$V_{d+1}=0$. An ordered pair of linear transformations A, B from V to V   is called LR whenever there exists a decomposition mmlsi178" class="mathmlsrc">${\{V_i\}}_{i=0}^d$ of V that is lowered by A and raised by B  . In this case the decomposition mmlsi178" class="mathmlsrc">${\{V_i\}}_{i=0}^d$ is uniquely determined by A, B  ; we call it the mmlsi10" class="mathmlsrc">(A,B)$(A,B)$-decomposition of V. Consider a 3-tuple of linear transformations A, B, C from V to V such that any two of A, B, C form an LR pair on V. Such a 3-tuple is called an LR triple on V. Let 伪, 尾, 纬   be nonzero scalars in mmlsi177" class="mathmlsrc">F$\mathbb{F}$. The triple 伪A, 尾B, 纬C is an LR triple on V, said to be associated to A, B, C  . Let mmlsi178" class="mathmlsrc">${\{V_i\}}_{i=0}^d$ be a decomposition of V and let X be a linear transformation from V to V. We say X   is tridiagonal with respect to mmlsi178" class="mathmlsrc">${\{V_i\}}_{i=0}^d$ whenever mmlsi11" class="mathmlsrc">XV_i⊆V_i−1+V_i+V_i+1$X{V}_i\subseteq V_{i-1}+V_i+V_{i+1}$ for mmlsi179" class="mathmlsrc">0≤i≤d$0\le i\le d$. Let mmlsi12" class="mathmlsrc">X$\mathcal{X}$ be the vector space over mmlsi177" class="mathmlsrc">F$\mathbb{F}$ consisting of the linear transformations from V to V   that are tridiagonal with respect to the mmlsi10" class="mathmlsrc">(A,B)$(A,B)$ and mmlsi13" class="mathmlsrc">(B,C)$(B,C)$ and mmlsi14" class="mathmlsrc">(C,A)$(C,A)$ decompositions of V. There is a special class of LR triples, called q  -Weyl type. In the present paper, we find a basis of mmlsi12" class="mathmlsrc">X$\mathcal{X}$ for each LR triple that is not associated to an LR triple of q-Weyl type.