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Canonical bases via pairing monomials
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abstract
For any quantum group of finite ADE type, we prove a new formula for the standard bilinear form evaluated at monomials. Combining this with ideas from the Lusztig-Shoji algorithm, we obtain a new algorithm that computes the canonical basis. In type A, the algorithm also computes composition multiplicities of standard modules for the affine Hecke algebra of $\text{GL}_n$ and we explain how the algorithm can be extended to compute the dimensions of simple modules.
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Algorithm for computing canonical bases and foldings of quantum groups
For non-symmetric quantum groups of finite type, the canonical-to-PBW transition matrix is computable by a closed inner-product formula, and is determined exactly by the sigma-invariant data of the folded symmetric case.
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