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Canonical bases via pairing monomials

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arxiv 2308.16254 v1 pith:TQZPYKEO submitted 2023-08-30 math.RT math.QA

classification math.RTmath.QA
keywords algorithmcanonicalcomputesmodulesmonomialsstandardtypeaffine
verification ladder T0 review T1 audit T2 compute T3 formal
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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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  1. Algorithm for computing canonical bases and foldings of quantum groups

    math.QA 2025-06 conditional novelty 7.0 of 10

    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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