Pieri's formula for generalized Schur polynomials
Yasuhide Numata
Hokkaido University Department of Mathematics Kita 10, Nishi 8, Kita-Ku Sapporo Hokkaido 060-0810 Japan
DOI: 10.1007/s10801-006-0047-y
Abstract
Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutation relation of generalized Schur operators implies Pieri's formula for generalized Schur polynomials.
Pages: 27–45
Keywords: keywords Pieri formula; generarized Schur operators; Schur polynomials; Young diagrams; planar binary trees; differential posets; dual graphs; symmetric functions; quasi-symmetric polynomials
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References
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2. S. Fomin, “Generalized Robinson-Schensted-Knuth correspondence,” Zariski Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986), 156-175, 195 (Russian); English transl., J. Soviet Math. 41 (1988), 979-991.
3. S. Fomin, “Duality of graded graphs,” J. Algebraic Combin. 3 (1994), 357-404.
4. S. Fomin, “Schensted algorithms for dual graded graphs,” J. Algebraic Combin. 4 (1995), 5-45.
5. S. Fomin, “Schur operators and Knuth correspondences,” J. Combin. Theory Ser. A 72 (1995), 277-292.
6. T. Lam, “A combinatorial generalization of the Boson-Fermion correspondence,” Math. Res. Lett. 13(3) (2006), 377-329.
7. Y. Numata, “An example of generalized Schur operators involving planar binary trees,” preprint, arXiv:math.CO/0609376.
8. T. Roby, “Applications and extensions of Fomin's generalization of the Robinson-Schensted correspondence to differential posets,” Ph.D. thesis, M.I.T., 1991.
9. R. Stanley, “Differential posets, J. Amer. Math. Soc., 1 (1988), 919-961.
10. R. Stanley, “Variations on differential posets,” in D. Stanton, (ed.), Invariant Theory and Tableaux IMA volumes in mathematics and its applications, Springer-Verlag, New York, pp. 145-165.