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Given the famous Littlewood-Richardson rule, in terms of Schur polynomials:

$$s_\mu s_\nu=\sum_\lambda c^{\lambda}_{\mu\nu} s_\lambda,$$

is there a classification of the cases where the LR coefficients are equal to 1?

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The answer is Yes, but this requires some elaboration.

  1. Knutson-Tao-Woodward prove Fulton's conjecture in $\S$6.1. In principle, you can follow the approach by De Loera-McAllister or Mulmuley-Narayanan-Sohoni, to convert the result into a statement about certain polytopes being single points.

  2. Bürgisser-Ikenmeyer make the answer much more explicit, see Prop. 3.13. Ikenmeyer further generalized this in to $c^\lambda_{\mu\nu}>t$ in his thesis, see Theorem 11.3.2.

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