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• W2153399488 abstract "Let $F$ denote a field and let $V$ denote a vector space over $F$ with finite positive dimension. We consider a pair of linear transformations $A:V to V$ and $A^*:V to V$ that satisfy the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering ${V_i}_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 leq i leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering ${V^*_i}_{i=0}^delta$ of the eigenspaces of $A^*$ such that $A V^*_i subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 leq i leq delta$, where $V^*_{-1}=0$ and $V^*_{delta+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW subseteq W$, $A^* W subseteq W$, $W neq 0$, $W neq V$. We call such a pair a {it tridiagonal pair} on $V$. It is known that $d=delta$ and for $0 leq i leq d$ the dimensions of $V_i$, $V_{d-i}$, $V^*_i$, $V^*_{d-i}$ coincide. We say the pair $A,A^*$ is {it sharp} whenever $dim V_0=1$. It is known that if $F$ is algebraically closed then $A,A^*$ is sharp. A conjectured classification of the sharp tridiagonal pairs was recently introduced by T. Ito and the second author. We present a result which supports the conjecture. Given scalars ${th_i}_{i=0}^d$, ${th^*_i}_{i=0}^d$ in $F$ that satisfy the known constraints on the eigenvalues of a tridiagonal pair, we define an $F$-algebra $T$ by generators and relations. We consider the algebra $e^*_0Te^*_0$ for a certain idempotent $e^*_0 in T$. Let $R$ denote the polynomial algebra over $F$ involving $d$ variables.We display a surjective algebra homomorphism $mu: R to e^*_0Te^*_0$. We conjecture that $mu$ is an isomorphism. We show that this $mu$-conjecture implies the classification conjecture, and that the $mu$-conjecture holds for $dleq 5$." @default.
• W2153399488 created "2016-06-24" @default.
• W2153399488 creator A5059249005 @default.
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• W2153399488 date "2009-08-18" @default.
• W2153399488 modified "2023-09-27" @default.
• W2153399488 title "Tridiagonal pairs and the $mu$-conjecture" @default.
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