Abstract:In order to give a combinatorial descriptions of tensor product multiplicites for semisimple groups, it is useful to find bases for representations which are compatible with the actions of Chevalley generators of the Lie algebra. There are three known examples of such bases, each of which flows from geometric or algebraic mountain. Remarkably, each mountain gives the same combinatorial shadow: the crystal B(∞) and the Mirković-Vilonen polytopes. In order to distinguish between the three bases, we introduce mea… Show more
“…, n}. One is interested in constructing bases of L n (λ) with good symmetry and positivity properties and with basis elements indexed by SSYT(λ, [n]) in a natural way, see [Kam22] for a recent survey. Lusztig and Kashiwara introduced the dual canonical basis (henceforth the "canonical" basis) as one solution to this problem.…”
We show that Lusztig's canonical basis for the degree two part of the Grassmannian coordinate ring is given by SL k web diagrams. Equivalently, we show that every SL 2 web immanant of a plabic graph for Gr(k, n) is an SL k web invariant.
“…, n}. One is interested in constructing bases of L n (λ) with good symmetry and positivity properties and with basis elements indexed by SSYT(λ, [n]) in a natural way, see [Kam22] for a recent survey. Lusztig and Kashiwara introduced the dual canonical basis (henceforth the "canonical" basis) as one solution to this problem.…”
We show that Lusztig's canonical basis for the degree two part of the Grassmannian coordinate ring is given by SL k web diagrams. Equivalently, we show that every SL 2 web immanant of a plabic graph for Gr(k, n) is an SL k web invariant.
“…, n}. One is interested in constructing bases of L n (λ) with good symmetry and positivity properties and with basis elements indexed by SSYT(λ, [n]) in a natural way, see [12] for a recent survey. Lusztig and Kashiwara introduced the dual canonical basis (henceforth the "canonical" basis) as one solution to this problem.…”
We show that Lusztig's canonical basis for the degree two part of the Grassmannian coordinate ring is given by SL(k) web diagrams. Equivalently, we show that every SL(2) web immanant of a plabic graph for Gr(k,n) is an SL(k) web invariant.
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