Bijective proofs of shifted tableau and alternating sign matrix identities
A.M. Hamel1
and R.C. King2
1Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada
2School of Mathematics, University of Southampton, Southampton SO17 1BJ, England
2School of Mathematics, University of Southampton, Southampton SO17 1BJ, England
DOI: 10.1007/s10801-006-0044-1
Abstract
We give a bijective proof of an identity relating primed shifted gl( n)-standard tableaux to the product of a gl( n) character in the form of a Schur function and Õ 1 \sterling i < j \sterling n ( x i + y j) {\prod_{1\leq i < j \leq n} (x_i + y_j)}. This result generalises a number of well-known results due to Robbins and Rumsey, Chapman, Tokuyama, Okada and Macdonald. An analogous result is then obtained in the case of primed shifted sp(2 n)-standard tableaux which are bijectively related to the product of a t-deformed sp(2 n) character and Õ 1 \sterling i < j \sterling n( x i+ t 2 x i -1+ y j+ t 2 y j -1) {\prod_{1\leq i < j \leq n}(x_i+t^2x_i^{-1}+y_j+t^2y_j^{-1})}. All results are also interpreted in terms of alternating sign matrix (ASM) identities, including a result regarding subsets of ASMs specified by conditions on certain restricted column sums.
Pages: 417–458
Keywords: keywords alternating sign matrices; shifted tableaux; Schur P-functions
Full Text: PDF
References
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2. D.M. Bressoud, “Three alternating sign matrix identities in search of bijective proofs,” Adv. Appl. Math. 27 (2001), 289-297.
3. R. Chapman, “Alternating sign matrices and tournaments,” Adv. Appl. Math. 27 (2001), 318-335.
4. A.M. Hamel and R.C. King, “Symplectic shifted tableaux and deformations of Weyl's denominator formula for sp(2n),” J. Algebraic Comb. 16 (2002), 269-300.
5. A.M. Hamel and R.C. King, “U-turn alternating sign matrices, symplectic shifted tableaux, and their weighted enumeration,” J. Algebraic Comb. 21 (2005), 395-421.
6. P.N. Hoffman and J.F. Humphreys, Projective Representations of the Symmetric Groups: Q-Functions and Shifted Tableaux, Oxford University Press, Oxford 1992.
7. G. Kuperberg, “Symmetry classes of alternating sign matrices under one roof,” Ann. of Math. (2) 156 (2002), 835-866.
8. I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd. ed. Oxford Univerity Press, Oxford 1995.
9. W.H. Mills, D.P. Robbins, and H. Rumsey, “Alternating sign matrices and descending plane partitions,” J. Comb. Theory A 34 (1983), 340-359.
10. S. Okada, “Partially strict shifted plane partitions,” J. Comb. Theory A 53 (1990), 143-156.
11. S. Okada, “Alternating sign matrices and some deformations of Weyl's denominator formula,” J. Algebraic Comb. 2 (1993), 155-176.
12. D.P. Robbins and H. Rumsey, “Determinants and alternating sign matrices,” Adv. Math. 62 (1986), 169-184.
13. B.E. Sagan, “Shifted tableaux, Schur Q-functions and a conjecture of Stanley,” J. Comb. Theory A 45 (1987), 62-103.
14. B.E. Sagan and R.P. Stanley, “Robinson Schensted algorithms for skew tableaux,” J. Comb.Theory A 55 (1990), 161-193.
15. T. Simpson, “Three generalizations of Weyl's denominator formula,” Elect. J. Comb. 3 (1997), # R12.
16. T. Simpson, “Another deformation of Weyl's denominator formula,” J. Comb. Theory A 77 (1997), 349-356.
17. T. Tokuyama, “A generating function of strict Gelfand patterns and some formulas on characters of general linear groups,” J. Math. Soc. Japan 40 (1988), 671-685.
18. D.R. Worley, “A theory of shifted Young tableaux,” Ph.D. thesis, M.I.T., 1984.
19. D. Zeilberger, “A proof of the alternating sign matrix conjecture,” Elect. J. Comb. 3 (1996), # R13.