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Tensors of rank $pi$

SPEAKER: Pavel Etingof

TITLE: Tensors of rank $pi$

ABSTRACT: Let $t$ be a complex number, and $V$ a complex vector space. I will explain how to define the tensor power $V^{otimes t}$. This can be done canonically if we fix a nonzero vector in $V$. However, the result is not a vector space but rather an (ind-)object in the tensor category ${rm Rep}(S_t)$, defined by P. Deligne as an interpolation of the representation category of the symmetric group $S_n$ to complex values of $n$. This category is semisimple abelian for $tnotin bold Z_+$, but only Karoubian (=idempotent complete) for $tin bold Z_+$, in which case it projects onto the usual representation category of $S_n$. I will recall the definition of the category ${rm Rep}(S_t)$, and explain how Schur-Weyl duality works in this category when $t notin bold Z_+$. If time permits, I will explain what happens at integer $t$, which is more subtle and is due to Inna Entova-Aizenbud.