Nonzero Symmetry Classes of Smallest Dimension

Chan, G. H. ; Lim, M. H.

Canadian journal of mathematics, Tome 32 (1980) no. 4, pp. 957-968

Résumé

Let U be a k-dimensional vector space over the complex numbers. Let ⊗m U denote the mth tensor power of U where m ≧ 2. For each permutation σ in the symmetric group Sm, there exists a linear mapping P(σ) on ⊗mU such that for all x 1, ..., xm in U.Let G be a subgroup of Sm and λ an irreducible (complex) character on G. The symmetrizer is a projection of ⊗ mU. Its range is denoted by Uλm(G) or simply Uλ(G) and is called the symmetry class of tensors corresponding to G and λ.

Détail

DOI : 10.4153/CJM-1980-073-0

Bibliographie

[1] 1. Boerner, H., Representations of groups (North Holland/American Elsevier, Amsterdam-London/New York, 1970). Google Scholar

[2] 2. Brauer, R., Investigations on group characters, Ann. of Math. 42 (1941), 926–985. Google Scholar

[3] 3. Burnside, W., Theory of groups of finite order (Reprinted by Dover, New York, 1955). Google Scholar

[4] 4. Chan, G. H., On a class of permutation groups, Nanta Math. 6 (1973), 100–105. Google Scholar

[5] 5. Chan, G. H., Construction of minimal (k)-groups, Nanta Math. 7 (1974), 30–35. Google Scholar

[6] 6. Chan, G. H., A characterization of minimal (k)-groups of degree n ≧ 3k, Linear and Multilinear Algebra 4 (1977), 285–305. Google Scholar

[7] 7. Chan, G. H., On the triviality of symmetry class of tensors, Linear and Multilinear Algebra 6 (1978), 73–82. Google Scholar

[8] 8. Chan, G. H., (k)-character and the triviality of symmetry classes, Linear Algebra Appl. (to appear). Google Scholar | DOI

[9] 9. Chan, G. H., A note on symmetrizers of rank one, Nanta Math. 11 (1979), 130–133. Google Scholar

[10] 10. Chang, S. C., On the vanishing of a (G,a) space, Chinese J. Math. 4 (1976), 1–7. Google Scholar

[11] 11. Curtis, U. C. and Reiner, I., Representation theory of finite groups and associative algebras (Interscience, New York-London, 1962). Google Scholar

[12] 12. Dieudonné, J. and Carrell, J. B., Invariant theory, old and new (Academic Press, New York, 1971). Google Scholar

[13] 13. Freese, R., Inequalities for generalized matrix functions based on arbitrary characters, Linear Algebra Appl. 7 (1973), 337–345. Google Scholar

[14] 14. Grone, R., A note on the dimension of an orbital subspace, Linear Algebra Appl. 17 (1977), 283–286. Google Scholar

[15] 15. Kerber, A., Representations of permutation groups I, Lecture Notes in Mathematics 240 (Springer-Verlag, New York, 1971). Google Scholar | DOI

[15] 15. Lim, M. H., Regular symmetry classes of tensors, Nanta Math. 8 (1975), 42–46. Google Scholar

[17] 17. Marcus, M., Finite dimensional multilinear algebra, Part 1 (Marcel Dekker, New-York, 1973). Google Scholar

[18] 18. Marcus, M., Finite dimensional multilinear algebra, Part II (Marcel Dekker, New York, 1975). Google Scholar

[19] 19. Merris, R., The dimensions of certain symmetry classes of tensors II, Linear and Multilinear Algebra 4 (1976), 205–207.' Google Scholar

[20] 20. Merris, R., Relations among generalized matrix functions, Pacific J. Math. 62 (1970), 153–161. Google Scholar

[21] 21. Merris, R., Nonzero decomposable symmetrized tensors, Linear Algebra Appl. 17 (1977), 287–292. Google Scholar

[22] 22. Merris, R., On vanishing decomposable symmetrized tensors, Linear and Multilinear Algebra 5 (1977), 79–86. Google Scholar

[23] 23. Merris, R., Recent advances in symmetry classes of tensors, preprint. Google Scholar | DOI

[24] 24. Merris, R. and Pierce, S., Elementary divisors of higher degree associated transformations, Linear and Multilinear Algebra 1 (1973), 241–250. Google Scholar

[25] 25. Merris, R. and Rashid, M. A., The dimensions of certain symmetry classes of tensors, Linear and Multilinear Algebra 2 (1974), 245–248. Google Scholar

[26] 26. Singh, K., On the vanishing of a pure product in a (G,a) space, Ph.D. Thesis, University of British Columbia (1967). Google Scholar

[27] 27. Westwick, R., A note on symmetry classes of tensors, J. Algebra, 15 (1970), 309–311. Google Scholar

[28] 28. Williamson, S. G., Symmetry operators of Kranz products, J. Comb. Theory 11 (1971), 122–138. Google Scholar