COMMENT ON ‘A DECOMPOSITION OF SCHUR FUNCTIONS AND AN ANALOGUE OF THE ROBINSON-SCHENSTED-KNUTH ALGORITHM’

S. MASON

The purpose of this comment is to clarify the connections between Demazure characters and the objects studied in this work. The nonsymmetric Macdonald polynomials introduced by Macdonald [7] and studied by Cherednick [1] are denoted by E_{α}(X; q, t), where α is a weak composition and X = (x_{1}, x_{2}, …). The Demazure characters introduced by Demazure in [2] and studied by Ion [3], Joseph [4], and Sanderson [10] are the specializations E_{α}(X; 0, 0).

Marshall [8] works with a variation of the above nonsymmetric polynomials obtained by reversing the indexing composition, reversing the variables, and replacing q and t by q^{−1} and t^{−1} respectively. These nonsymmetric polynomials, denoted Ê_{α}(X; q, t), can therefore be written as Ê_{α}(x_{1}, x_{2}, …; q, t) = E_{reverse(α)}(…, x_{2}, x_{1}; q^{−1}, t^{−1}). It is these polynomials that we specialize to obtain the polynomials explored in this paper. In fact, the specializations of the Ê_{α}(X; q, t) to q = t = 0 are equivalent to the second family of Demazure characters, often called “standard bases” or “Demazure atoms”, introduced by Lascoux and Schützenberger in [5] and studied by Lascoux in [6]. Please see [9] for a combinatorial proof of this equivalence.

We provide the following short table for the partition λ = (2, 1, 0) to illustrate the distinction between E_{α}(X; 0, 0) and Ê_{α}(X; 0, 0).

Composition α  E_{α}(X; 0, 0)  Ê_{α}(X; 0, 0)
(2, 1, 0)  x_{1}^{2}x_{2}  x_{1}^{2}x_{2}
(2, 0, 1)  x_{1}^{2}x_{2} + x_{1}^{2}x_{3}  x_{1}^{2}x_{3}
(1, 2, 0)  x_{1}^{2}x_{2} + x_{1}x_{2}^{2}  x_{1}x_{2}^{2}
(1, 0, 2)  x_{1}^{2}x_{2} + x_{1}x_{2}^{2} + x_{1}^{2}x_{3} + x_{1}x_{2}x_{3} + x_{1}x_{3}^{2}  x_{1}x_{2}x_{3} + x_{1}x_{3}^{2}
(0, 2, 1)  x_{1}^{2}x_{2} + x_{1}x_{2}^{2} + x_{1}^{2}x_{3} + x_{1}x_{2}x_{3} + x_{2}^{2}x_{3}  x_{1}x_{2}x_{3} + x_{2}^{2}x_{3}
(0, 1, 2)  x_{1}^{2}x_{2} + x_{1}x_{2}^{2} + x_{1}^{2}x_{3} + 2x_{1}x_{2}x_{3} + x_{2}^{2}x_{3} + x_{1}x_{3}^{2} + x_{2}x_{3}^{2}  x_{2}x_{3}^{2}

References

[1] Cherednik, I,. Nonsymmetric Macdonald polynomials, Math. Res. Notices, 10 (1995), pp. 483–515.

[2] Demazure, M., Désingularisation des variétés de Schubert, Ann. E. N. S., 6 (1974), 163–172.

[3] Ion, Bogdan, Nonsymmetric Macdonald polynomials and Demazure characters, Duke Mathematical Journal, 116 (2003), 299–318

[4] Joseph, A., On the Demazure character formula, Ann. Sci. École Norm. Sup. (4), 18:3 (1985), 389–419.

[5] Lascoux, A., and Schützenberger, M.-P., Keys and Standard Bases, Invariant Theory and Tableaux, IMA Volumes in Math and its Applications (D. Stanton, ED.), Southend on Sea, UK, 19 (1990), 125–144.

[6] A. Lascoux, Double Crystal graphs, Studies in Memory of Issai Schur, Progress In Math. 210, Birkhaüser (2003) 95–114.

[7] Macdonald, I. G., Affine Hecke algebras and orthogonal polynomials, Astérisque 237 (1996), pp.189–207, Séminaire Bourbaki 1994/95, Exp. no. 797.

[8] Dan Marshall, Symmetric and nonsymmetric Macdonald polynomials. On combinatorics and statistical mechanics, Ann. Comb. 3 (1999), no. 2-4, 385–415.

[9] Mason, S., An explicit construction of type A Demazure Atoms, to appear in J. Algebraic Combinatorics.

[10] Sanderson, Y., On the Connection between Macdonald polynomials and Demazure characters, J. Algebraic Combin. 11 (2000), no.3, 269-275.

Department of Mathematics, Davidson College

E-mail address: samason@davidson.edu
URL: http://www.davidson.edu/math/mason