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 ![{lambda}](http://jlms.oxfordjournals.org/math/lambda.gif) denotes the associated irreducible character of Sn,the symmetric group on {1, 2, ..., n}, and, if c CSn, 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 c CSn 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 (![{lambda}](http://jlms.oxfordjournals.org/math/lambda.gif) , {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 ![Formula](http://jlms.oxfordjournals.org/content/vol49/issue1/fulltext/40/f4.gif) 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 ![Formula](http://jlms.oxfordjournals.org/content/vol49/issue1/fulltext/40/f4.gif) ß and ß ![Formula](http://jlms.oxfordjournals.org/content/vol49/issue1/fulltext/40/f4.gif) are both false. |
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