Geodesic
Interpolation analogues of Schur $Q$-functions
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part X, Tome 307 (2004), pp. 99-119 Cet article a éte moissonné depuis la source Math-Net.Ru

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We introduce interpolation analogues of the Schur $Q$-functions – the multiparameter Schur $Q$-functions. We obtain for them several results: a combinatorial formula, generating functions for one-row and two-rows functions, vanishing and characterization properties, a Pieri-type formula, a Nimmo-type formula (a relation of two Pfaffians), a Giambelli–Schur-type Pfaffian formula, a determinantal formula for the transition coefficients between multiparameter Schur $Q$-functions with different parameters. We write an explicit Pfaffian expression for the dimension of a skew shifted Young diagram. This paper is a continuation of the author's paper math.CO/0303169 and is a partial projective analogue of the paper q-alg/9605042 by A. Okounkov and G. Olshanski, and of the paper math.CO/0110077 by G. Olshanski, A. Regev, and A. Vershik. 
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     title = {Interpolation analogues of {Schur} $Q$-functions},
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V. N. Ivanov. Interpolation analogues of Schur $Q$-functions. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part X, Tome 307 (2004), pp. 99-119. http://geodesic.mathdoc.fr/item/ZNSL_2004_307_a2/

[1] L. Biedenharn, J. Louck, “A new class of symmetric polynomials defined in terms of tableaux”, Adv. in Appl. Math., 10 (1989), 396–438 | DOI | MR | Zbl

[2] S. Billey, M. Haiman, “Schubert polynomials for the classical groups”, J. Amer. Math. Soc., 8:2 (1995), 443–482 | DOI | MR | Zbl

[3] A. Borodin, G. Olshanski, “Harmonic functions on multiplicative graphs and interpolation polynomials”, Electron. J. Combin., 7 (2000), R28 ; arXiv: /math.CO/9912124 | MR | Zbl

[4] J. Math. Sci., 96:5 (1999), 3472–3477 | DOI | MR | Zbl

[5] S. Fomin, A. N. Kirillov, “Combinatorial $B_n$-analogues of Schubert polynomials”, Trans. Amer. Math. Soc., 348 (1996), 3591–3620 ; arXiv: /hep-th/9306093 | DOI | MR | Zbl

[6] P. N. Hoffman, J. F. Humphreys, Projective Representations of the Symmetric Groups, Oxford University Press, New York, 1992 | MR | Zbl

[7] V. N. Ivanov, “Dimension of a skew shifted Young diagram and projective representations of the infinite symmetric group”, Zap. Nauchn. Semin. POMI, 240, 1997, 115–135 ; J. Math. Sci., 96:5 (1999), 3517–3530 ; arXiv: /math.CO/0303169 | MR | Zbl | DOI

[8] J. Math. Sci., 107:5 (2001), 4195–4211 | DOI | MR | Zbl

[9] V. N. Ivanov, “Gaussian limit for projective characters of large symmetric groups”, Zap. Nauchn. Semin. POMI, 283, 2001, 73–97 | MR | Zbl

[10] T. Józefiak, “Characters of projective representations of symmetric groups”, Expo. Math., 7 (1989), 193–247 | MR | Zbl

[11] S. Kerov, A. Okounkov, G. Olshanski, “The boundary of Young graph with Jack edge multiplicities”, Internat. Math. Res. Notices, 4 (1998), 173–199 ; arXiv: /q-alg/9703037 | DOI | MR | Zbl

[12] F. Knop, “Symmetric and non-symmetric quantum Capelli polynomials”, Comment. Math. Helv., 72 (1997), 84–100 ; arXiv: /q-alg/9603028 | MR | Zbl

[13] F. Knop, S. Sahi, “Difference equations and symmetric polynomials defined by their zeros”, Internat. Math. Res. Notices, 10 (1996), 473–486 ; arXiv: /q-alg/9610017 | DOI | MR | Zbl

[14] A. Lascoux, “Puissances extérieurs, déterminants et cycles de Schubert”, Bull. Soc. Math. France, 102 (1974), 161–179 | MR | Zbl

[15] A. Lascoux, Notes on interpolation in one and several variables, http://phalanstere.univ-mlv.fr/

[16] A. Lascoux, P. Pragacz, “Operator calculus for Q-polynomials and Schubert polynomials”, Adv. Math., 140 (1998), 1–43 | DOI | MR | Zbl

[17] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition (with contributions by A. Zelevinsky), The Clarendon Press, Oxford University Press, New York, 1995 | MR | Zbl

[18] A. Molev, “Factorial supersymmetric Schur functions and super Capelli identities”, Kirillov's Seminar on Representation Theory, Amer. Math. Soc. Transl., Ser. 2, 181, ed. G. I. Olshanski, Amer. Math. Soc., Providence, RI, 1998, 109–137 ; arXiv: /q-alg/9606008 | MR | Zbl

[19] M. L. Nazarov, “Projective representations of the infinite symmetric group”, Representation Theory and Dynamical Systems, Adv. Soviet Math., 9, ed. A. M. Vershik, Amer. Math. Soc., Providence, RI, 1992, 115–130 | MR

[20] M. Nazarov, “Capelli identities for Lie superalgebras”, Ann. Sci. École. Norm. Sup., 30 (1997), 847–872 ; arXiv: /q-alg/9610032 | MR | Zbl

[21] J. J. C. Nimmo, “Hall–Littlewood symmetric functions and the BKP equation”, J. Phys. A, 23 (1990), 751–760 | DOI | MR | Zbl

[22] A. Yu. Okounkov, “Quantum immanants and higher Capelli identities”, Transformation Groups, 1 (1996), 99–126 ; arXiv: /q-alg/9602028 | DOI | MR | Zbl

[23] A. Okounkov, “(Shifted) Macdonald polynomials: $q$-integral representation and combinatorial formula”, Compositio Math., 112:2 (1998), 147–182 ; arXiv: /q-alg/9605013 | DOI | MR | Zbl

[24] A. Okounkov, “A characterization of interpolation Macdonald polynomials”, Adv. in Appl. Math., 20:4 (1998), 395–428 ; arXiv: /q-alg/9712052 | DOI | MR | Zbl

[25] A. Okounkov, G. Olshanski, “Shifted Schur functions”, Algebra i Analiz, 9:2 (1997), 73–146 ; St. Petersburg Math. J., 9 (1998), 239–300 ; arXiv: /q-alg/9605042 | MR | Zbl

[26] A. Okounkov, G. Olshanski, “Shifted Schur functions. II: The binomial formula for characters of classical groups and its applications”, Kirillov's Seminar on Representation Theory, Amer. Math. Soc. Transl., Ser. 2, 181, ed. G. I. Olshanski, Amer. Math. Soc., Providence, RI, 1998, 245–271 ; arXiv: /q-alg/9612025 | MR | Zbl

[27] A. Okounkov, G. Olshanski, “Shifted Jack polynomials, binomial formula, and applications”, Math. Res. Lett., 4:1 (1997), 69–78 ; arXiv: /q-alg/9608020 | MR | Zbl

[28] G. Olshanski, A. Regev, A. Vershik, “Frobenius–Schur functions”, Studies in memory of Issai Schur, Progr. Math., 210, eds. A. Joseph, A. Melnikov, R. Rentschler, Birkhäuser, 2003, 251–299 ; arXiv: /math.CO/0110077 | MR | Zbl

[29] P. Pragacz, “Algebro-geometric applications of Schur $S$- and $Q$-polynomials”, Lect. Notes Math., 1478, 1991, 130–191 | MR | Zbl

[30] S. Sahi, “Interpolation, integrality, and a generalization of Macdonald's polynomials”, Internat. Math. Res. Notices, 1996, no. 10, 457–471 | DOI | MR | Zbl

[31] I. Schur, “Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrocheme lineare Substitionen”, J. Reine Angew. Math., 139 (1911), 155–250 | DOI | Zbl

[32] A. N. Sergeev, “Tensor algebra of the identity representation as a module over Lie superalgebras GL(n,m) and Q(n)”, Mat. Sb., 123 (1984), 422–430 | MR

[33] J. R. Stembridge, “Shifted tableaux and the projective representations of symmetric groups”, Adv. Math., 74 (1989), 87–134 | DOI | MR | Zbl

[34] J. R. Stembridge, “Nonintersecting paths, Pfaffians, and plane partitions”, Adv. Math., 83 (1990), 96–131 | DOI | MR

[35] M. Yamaguchi, “A duality of the twisted group algebra of the symmetric group and a Lie superalgebra”, J. Algebra, 222:1 (1999), 301–327 ; arXiv: /math.RT/9811090 | DOI | MR | Zbl

[36] H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton Univ. Press, Princeton, NJ, 1939 | MR