Geometric complexity theory III: on deciding nonvanishing of a Littlewood---Richardson coefficient

  • Authors:

  • Ketan D. Mulmuley;Hariharan Narayanan;Milind Sohoni

  • Affiliations:

  • Department of Computer Science, University of Chicago, Chicago, USA 60637;Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, USA 02139;Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Mumbai, India 400 076

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 2012

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Abstract

We point out that the positivity of a Littlewood---Richardson coefficient $c^{\gamma}_{\alpha, \beta}$ for sl n can be decided in strongly polynomial time. This means that the number of arithmetic operations is polynomial in n and independent of the bit lengths of the specifications of the partitions 驴,β, and 驴, and each operation involves numbers whose bitlength is polynomial in n and the bit lengths 驴,β, and 驴.Secondly, we observe that nonvanishing of a generalized Littlewood---Richardson coefficient of any type can be decided in strongly polynomial time assuming an analogue of the saturation conjecture for these types, and that for weights 驴,β,驴, the positivity of $c^{ 2\gamma}_{2\alpha, 2\beta}$ can (unconditionally) be decided in strongly polynomial time.