Immanant Inequalities, Induced Characters, and Rank Two Partitions |
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Authors: | Pate Thomas H |
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Institution: | Mathematics Department, Parker Hall, Auburn University Auburn, Alabama 36849, USA |
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Abstract: | If = {1, 2, ..., s}, where 1 2 ... s > 0, is a partitionof n then denotes the associated irreducible character of Sn,the symmetric group on {1, 2, ..., n}, and, if cCSn, the groupalgebra generated by C and Sn, then dc(·) denotes thegeneralized matrix function associated with c. If c1, c2 CSnthen we write c1 c2 in case (A) (A) for each n x n positivesemi-definite Hermitian matrix A. If cCSn and c(e) 0, wheree denotes the identity in Sn, then or denotes (c(e))–1 c. The main result, an estimate for the norms of tensors of a certainanti-symmetry type, implies that if = {1, 2, ..., s, 1t} isa partition of n such that s > 1 and s = 2, and ' denotes{1, 2, ..., s-1, 1t} then (, {2}) where denotes characterinduction from Sn–2 x S2 to Sn. This in turn implies thatif = {1, 2, ..., s, 1t} with s > 1, s = 2, and ßdenotes {1 + 2, 2, ..., s-1, 1t} then ß which,in conjunction with other known results, provides many new inequalitiesamong immanants. In particular it implies that the permanentfunction dominates all normalized immanants whose associatedpartitions are of rank 2, a result which has proved elusivefor some years. We also consider the non-relationship problem for immanants– that is the problem of identifying pairs, (,ß)such that ß and ß are both false. |
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