Factorisation of Littlewood-Richardson coefficients. (English) Zbl 1207.05214
Summary: The hive model is used to show that the saturation of any essential Horn inequality leads to the factorisation of Littlewood-Richardson coefficients. The proof is based on the use of combinatorial objects known as puzzles. These are shown not only to account for the origin of Horn inequalities, but also to determine the constraints on hives that lead to factorisation. Defining a primitive Littlewood-Richardson coefficient to be one for which all essential Horn inequalities are strict, it is shown that every Littlewood-Richardson coefficient can be expressed as a product of primitive coefficients. Precisely the same result is shown to apply to the polynomials defined by stretched Littlewood-Richardson coefficients.
MSC:
| 05E05 | Symmetric functions and generalizations |
| 05E10 | Combinatorial aspects of representation theory |
Keywords:
Littlewood-Richardson coefficients; hive model; Horn inequalities; puzzles; factorisation; LR-polynomialsCitations:
Zbl 0915.14010; Zbl 0944.05097; Zbl 0873.14016; Zbl 0815.15012; Zbl 0935.05093; Zbl 0929.15006; Zbl 0994.15021; Zbl 0112.01501; Zbl 1042.14031; Zbl 1043.05111; Zbl 1178.05101; Zbl 1178.05100; Zbl 1071.15019; Zbl 0979.20041; Zbl 1124.05092; Zbl 1156.05061; Zbl 1018.16012References:
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