JOURNAL OF
ALGEBRAIC
COMBINATORICS

s _{[( l)\tilde]} (1, q, \frac{1}{4} , q ⁿ ) - q ⁿ s _{[( l)\tilde]} (1, q, \frac{1}{4} , q ⁿ ) s_{\tilde λ} (1,q, \ldots ,q^n ) - q^n s_{\tilde λ} (1,q, \ldots ,q^n )

1. M. Aigner, “Lexicographic matching in Boolean algebras,” J. Combin. Th. B 14 (1973), 187-194.
2. K. Akin, D. Buchsbaum, and J. Weyman, “Schur functors and Schur complexes,” Adv. Math. 44 (1982), 207-278.
3. A. Berele, “Construction of Sp-modules by tableaux,” Lin. and Multin. Alg. 19 (1986), 299-307.
4. N. Bourbaki, Groups et alg`ebres de lie, Masson, Paris (1990).
5. G.E. Andrews, “On the differences of successive Gaussian polynomials,” J. Stat. Plan. Inf. 34 (1993), 19-22.
6. L. Butler, “A unimodality result in the enumeration of subgroups of a finite abelian group,” Proc. Amer. Math. Soc. 101 (1987), 771-775.
7. C. DeConcini, “Symplectic standard tableaux,” Adv. Math. 34 (1979), 1-27.
8. R. Donnelly, Personal communication, 1995.
9. S. Fishel, “Nonnegativity results for generalized q-binomial coefficients,” Disc. Math. 147 (1995), 121-137.
10. W. Fulton and J. Harris, “Representation theory: A first course,” Graduate Texts in Mathematics, Springer- Verlag, New York, 1991, Vol. 129.
11. C. Greene and D. Kleitman, “Proof techniques in the theory of finite sets,” Studies in Combinatorics, G.-C. Rota (Ed.), Mathematical Association of America, 1978, pp. 22-79.
12. J. Hughes, “Lie algebraic proofs of some theorems on partitions,” Number Theory and Algebra, H. Zassenhaus (Ed.), Academic Press, New York, 1977.
13. W.M. Kantor, “On incidence matrices of finite projective and affine spaces,” Math. Zeit. 124 (1972), 315-318. P1: PMR Journal of Algebraic Combinatorics KL507-06-Reiner November 7, 1997 9:20 UNIMODALITY OF DIFFERENCES OF SPECIALIZED SCHUR FUNCTIONS 107
14. R.C. King, “Weight multiplicities for the classical groups,” Lecture Notes in Physics, Springer-Verlag, New York, 1975, Vol. 50.
15. A. Kirillov, “Unimodality of generalized Gaussian coefficients,” C.R. Acad. Sci. Paris Ser. I 315 (1992), 497-501.
16. K. Koike and I. Terada, “Young diagrammatic methods for the representation theory of the classical groups of type Bn, Cn, Dn,” J. Algebra 107 (1987), 466-511.
17. D.E. Littlewood, Theory of Group Characters, Oxford University Press, Oxford, 1940.
18. I. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, New York, 1979.
19. M. Maliakas and P. Olver, “Explicit generalized Pieri maps,” J. Algebra 148 (1992), 68-85.
20. R. Proctor, “Representations of sl(2, C) on posets and the Sperner property,” SIAM J. Algebraic Discrete Math. 3 (1982), 275-280. 21. ---, “Solution of a Sperner conjecture of Stanley with a construction of Gelfand,” J. Comb. Th. A 54 (1990), 225-234. 22. ---, “Bruhat lattices, plane partition generating functions, and minuscule representations,” Europ. J. Combin. 5 (1984), 331-350. 23. ---, “Young tableaux, Gelfand patterns, and branching rules for classical groups,” J. Algebra 164 (1994), 299-360.
24. J. Sheats, “A symplectic jeu-de-taquin bijection between the tableaux of King and of DeConcini,” Preprint 1995.
25. R. Stanley, “Log-concave and unimodal sequences in algebra, combinatorics, and geometry,” Annals of the New York Academy of Sciences 576 (1989), 500-535.
26. S. Sundaram, “Orthogonal tableaux and an insertion algorithm for S O(2n + 1),” J. Comb. Th. A 53 (1990), 239-256.
27. D. White and S.G. Williamson, “Recursive matching algorithms and linear orders on the subset lattice,” J. Combin. Th. A 23 (1977), 117-127.
28. D. Zeilberger, “A one-line high school algebra proof of the unimodality of the Gaussian polynomials [ n ] for k k < 20,” in q-Series and Partitions, D. Stanton (Ed.), Springer, New York, 1989, pp. 67-72.