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Some Conjectures for Immanants

Published online by Cambridge University Press:  20 November 2018

J. R. Stembridge
J. R. Stembridge*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan, U.S.A. 48109-1003
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Abstract

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We present a series of conjectures for immanants, together with the supporting evidence we possess for them. The conjectures are loosely organized into three families. The first concerns inequalities involving the immanants of totally positive matrices (Le.,real matrices with nonnegative minors). This includes, for example, the conjecture that immanants of totally positive matrices are nonnegative. The second family involves the immanants of Jacobi-Trudi matrices. These conjectures were suggested by a previous conjecture of Goulden and Jackson (recently proved by C. Greene) that the immanants of Jacobi-Trudi matrices are polynomials with nonnegative coefficients. The third family involves geometric and combinatorial structures associated with total positivity and paths in acyclic digraphs.

Keywords

15A1505E0520C3015A45
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 5 , 01 October 1992 , pp. 1079 - 1099
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Curtis, C.W. and Reiner, I., Methods of Representation Theory, Vol I, Wiley, New York, 1981.Google Scholar
2. Eğecioğlu, O.N. and Remmel, J.B., The monomial symmetric functions and the Frobeniusmap, J. Combin. Theory (A) 54(1990), 272–295.Google Scholar
3. Gessel, I.M. and Viennot, G., Determinants, paths, and plane partitions, preprint.Google Scholar
4. R, I. Goulden and Jackson, D.M., Immanants of combinatorial matrices, J. Algebra 148(1992), 305–324.Google Scholar
5. Greene, C., Proof of a conjecture on immanants of the Jacobi-Trudi matrix, Linear Algebra Appl. 171(1992), 65–79.Google Scholar
6. Haiman, M.D., Immanant conjectures and Kazhdan-Lusztig polynomials, preprint.Google Scholar
7. Heyfron, P., Immanant dominance orderingsfor hook partitions, Linear and Multilinear Algebra 24(1988), 65–78.Google Scholar
8. James, G.D. and Kerber, A., The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, MA, 1981.Google Scholar
9. Johnson, C.R., private communication. Karlin, S., Total Positivity, Stanford Univ. Press, 1968.Google Scholar
11. Karlin, S., Coincident probabilities and applications in combinatorics, J. Applied Prob., (ed. Gani, J.), Supplementary Vol. 25A(1988), 185–200.Google Scholar
12. Karlin, S. and Rinott, Y., generalized Cauchy-Binet formula and applications to total positivity and majorization, J. Multivariate Anal. 27(1988), 284–299.Google Scholar
13. Koteljanski, D.M.ĭ, The theory of nonnegativeand oscillating matrices, Amer. Math. Soc. Transi. 27(1963), 1–8.Google Scholar
14. Lieb, E., Proofs of some conjectures on permanents, J. Math. Mech. 16(1966), 127–134.Google Scholar
15. Macdonald, I.G., Symmetric Functions and Hall Polynomials, Oxford Univ. Press, Oxford, 1979.Google Scholar
16. Merris, R., Single-hook characters and Hamiltonian circuits, Linear and Multilinear Algebra 14(1983), 21–35.Google Scholar
17. Schur, I., Über endliche Gruppen undHermitesche Formen, Math. Z. 1(1918), 184–207.Google Scholar
18. Stanley, R.P., Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, Monterey, 1986.Google Scholar
19. Stanley, R.P. and Stembridge, J.R., On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory (A), to appear.Google Scholar
20. Stembridge, J.R., Nonintersecting paths, pfaffians and plane partitions, Adv. in Math. 83(1990), 96–131.Google Scholar
21. Stembridge, J.R.,Immaniants of totally positive matrices are nonnegative, Bull. London Math. Soc. 23(1991), 422–428.Google Scholar