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BIRS Workshop Lecture Videos

Some $q$-exponential Formulas Involving the Double Lowering Operator $\psi$ for a Thin Tridiagonal Pair Bockting-Conrad, Sarah


Let $\mathbb{K}$ denote an algebraically closed field and let $V$ denote a vector space over $\mathbb{K}$ with finite positive dimension. In this talk, we consider a tridiagonal pair $A,A^*$ on $V$ which has $q$-Racah type. We will introduce the linear transformations $\psi:V\to V$, $\Delta:V \to V$, and $\mathcal{M}:V\to V$, each of which acts on the split decompositions of $V$ in an attractive way. We will show that $\Delta$ can be factored into a $q^{-1}$-exponential in $\psi$ times a $q$-exponential in $\psi$. We view $\Delta$ as a transition matrix from the first split decomposition of $V$ to the second. Consequently, we view the $q^{-1}$-exponential in $\psi$ as a transition matrix from the first split decomposition to a decomposition of $V$ which we interpret as a kind of half-way point. This half-way point turns out to be the eigenspace decomposition of $\mathcal{M}$. We will discuss the eigenspace decomposition of $\mathcal{M}$ and give the actions of various operators on this decomposition.

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