Abstract
We introduce and study a new basis in the algebra of symmetric functions. The elements of this basis are called the Frobenius—Schur functions (FS- functions, for short).
Our main motivation for studying the FS-functions is the fact that they enter a formula expressing the combinatorial dimension of a skew Young diagram in terms of the Frobenius coordinates. This formula plays a key role in the asymptotic character theory of the symmetric groups. The (FS-functions are inhomogeneous, and their top homogeneous components coincide with the conventional Schur functions (S-functions, for short). The (FS-functions are best described in the super realization of the algebra of symmetric functions. As supersymmetric functions, the(FS-functions can be characterized as a solution to an interpolation problem.
Our main result is a simple determinantal formula for the transition coefficients between theFS-and S-functions. We also establish theFSanalogs for a number of basic facts concerning the S-functions: the Jacobi—Trudi formula together with its dual form, the combinatorial formula (expression in terms of tableaux), the Giambelli formula and the Sergeev—Pragacz formula.
All these results hold for a large family of bases interpolating between theFS-functions and the ordinary S-functions.
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References
A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebrasAdv. Math.64 (1987), 118–175.
L. C. Biedenharn and J. D. Louck, A new class of symmetric polynomials defined in terms of tableauxAdvances in Appl. Math. 10(1989), 396–438.
L. C. Biedenharn and J. D. Louck, Inhomogeneous basis set of symmetric polynomials defined by tableauxProc. Nat. Acad. Sci. U.S.A.87 (1990), 1441–1445.
A. Borodin and G. Olshanski, Harmonic functions on multiplicative graphs and interpolation polynomialsElectronic J. Combinatorics7 (2000), #R28; math/9912124.
W. Y. C. Chen and J. D. Louck, The factorial Schur functionJ. Math. Phys.34 (1993), 4144–4160.
I. Gessel and G. Viennot, Binomial determinants, paths, and hook length formulaAdvances in Math.(1985), 300–321.
I. Goulden and C. Greene, A new tableau representation for supersymmetric Schur functionsJ. Algebra170 (1994), 687–703.
V. N. Ivanov, Dimension of skew shifted Young diagrams and projective repre-sentations of the infinite symmetric group. InRepresentation Theory Dynamical Systems Combinatorial and Algorithmical Methods II (A. M. Vershik, ed.), Zapiski Nauchnykh Seminarov POMI 240 Nauka, St. Petersburg, 1997, pp. 115–135 (Russian); English translation:J. Math. Sci.96, No. 5 (1999), 3517–3530.
V. N. Ivanov, Combinatorial formula for factorial Schur Q-functions. InRepre- sentation Theory Dynamical Systems Combinatorial and Algorithmical MethodsIII (A. M. Vershik, ed.), Zapiski Nauchnykh Seminarov POMI 256 Nauka, St. Petersburg, 1999, pp. 73–94 (Russian); English translation:J. Math. Sci. 107, No. 5 (2001), 4195–4211.
S. Kerov and G. Olshanski, Polynomial functions on the set of Young diagramsComptes Rendus Acad. Sci. Paris Sér. I319 (1994), 121–126.
S. Kerov, A. Okounkov, and G. Olshanski, The boundary of Young graph with Jack edge multiplicitiesIntern. Math. Res. NoticesNo. 4 (1998), 173–199.
S. Kerov and A. Vershik, The characters of the infinite symmetric group and probability properties of the Robinson—Schensted—Knuth algorithmSIAM J. Alg. Discr. Meth.7 (1986), 116–124.
S. Kerov and A. Vershik The Grothendieck group of the infinite symmetric group and symmetric functions with the elements of the Ko-functor theory of AF-algebras. InRepresentation of Lie Groups and Related TopicsAdv. Stud. Contemp. Math. 7 (A. M. VershikandD. P. Zhelobenko, eds.), Gordon and Breach, 1990, pp. 36–114.
A. Lascoux, Puissances extérieurs, déterminants et cycles de SchubertBull. Soc. Math. France102 (1974), 161–179.
A. Lascoux, Notes on interpolation in one and several variables, Preprint, available viahttp://phalanstere.univ-mlv.fr/-al/MAIN/publications.html
I. G. MacdonaldSymmetric Functions and Hall Polynomials2nd edition, University Press, 1995.
I. G. Macdonald, Schur functions: Theme and variations, Publ. I.R.M.A. Strasbourg, 498/S-27, Actes 28ième Séminaire Lotharingien (1992), 5–39.
I. G. MacdonaldNotes on Schubert PolynomialsPubl. LACIM, Universitédu Quebec, Montréal, 1991.
A. Molev, Factorial supersymmetric Schur functions and super Capelli identities. InKirillov’s Seminar on Representation Theory(G. Olshanski, ed.), American Mathematical Society Translations (2), Vol. 181, Amer. Math. Soc. Providence, RI, 1997, pp. 109–137.
A. I. Molev and B. E. Sagan, A Littlewood—Richardson rule for factorial Schur functionsTrans. Amer. Math. Soc.351 (1999), 4429–4443.
A. Okounkov, Quantum immanants and higher Capelli identitiesTransformation Groups 1(1996), 99–126.
A. Okounkov, On Newton interpolation of symmetric functions: A characterization of interpolation Macdonald polynomialsAdv. Appl. Math.20 (1998), 395–428.
A. Okounkov and G. Olshanski, Shifted Schur functionsAlgebra i Analiz9,No. 2 (1997), 73–146 (Russian): English translation:St. Petersburg Math. J.9 (1998), 239–300.
A. Okounkov and G. Olshanski, Shifted Jack polynomials, binomial formula, and applicationsMath. Research Letters4 (1997), 69–78.
G. Olshanski, Point processes and the infinite symmetric group. Part I: The general formalism and the density function, math/9804086.
P. Pragacz, Algebro-geometric applications of SchurS-and Q-polynomials. InTopics in Invariant TheorySeminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin, Lecture Notes in Math. 1478,Springer—Verlag, New York, Berlin, 1991, pp. 130–191.
P. Pragacz and A. Thorup, On a Jacobi—Trudi identity for supersymmetric polynomialsAdv. Math.(1992), 8–17.
A. Regev and T. Seeman, Shuffle invariance of the super-RSK algorithmAdvances in Appl. Math.to appear, math/0103206.
B. E. SaganThe Symmetric Group. Representations Combinatorial Algo-rithms and Symmetric FunctionsBrooks/Cole Publ. Co., Pacific Grove, CA, 1991.
J. R. Stembridge, A characterization of supersymmetric polynomialsJ. Alge-bra95, (1985), 439–444.
E. Thoma, Die unzerlegbaren, positive-definiten Klassenfunktionen der abzähl bar unendlichen, symmetrischen GruppeMath. Zeitschr85 (1964), 40–61.
A. M. Vershik and S. V. Kerov, Asymptotic theory of characters of the symmetric groupFunct. Anal. Appl.15,No. 4 (1981), 246–255.
A. J. Wassermann, Automorphic actions of compact groups on operator alge-bras, Thesis, University of Pennsylvania, 1981.
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Olshanski, G., Regev, A., Vershik, A., Ivanov, V. (2003). Frobenius-Schur Functions. In: Joseph, A., Melnikov, A., Rentschler, R. (eds) Studies in Memory of Issai Schur. Progress in Mathematics, vol 210. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0045-1_10
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