Computing Multiplicities of Lie Group Representations

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" summary

Algorithm computes branching multiplicities when group/homomorphism and number of diagram rows are fixed, running in poly-time for fixed rank (polynomial in log n). Method reduces highest-weight decomposition to weight multiplicities via Kostka-type partition functions, evaluates them with Barvinok lattice-point counting, then recovers highest-weight multiplicities by finite-difference inversion (derivative principle). Implementation (leadspeak.org) handles d=3 and extremely large box counts (reports up to 10^8), and recent refinements by Panova and Pak speed up zero/nonzero tests.

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" chapters (15)

1. 0:00Introduction
2. 1:07Talk structure
3. 3:13Branching problem setup
4. 5:26Kostka numbers example
5. 10:32Littlewood-Richardson example
6. 13:55Kronecker coefficients example
7. 16:00Plethysm coefficients example
8. 19:54Known results overview
9. 27:57Main theorem
10. 33:25Proof strategy
11. 36:35Weight multiplicities step
12. 43:06Branching decomposition step
13. 46:38Recovering multiplicities step
14. 50:08Kronecker simpler case
15. 54:09Software results

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