Ninth variation of classical group characters of type A-D and Littlewood identities. (English) Zbl 1532.05169
Summary: We introduce certain generalisations of the characters of the classical Lie groups, extending the recently defined factorial characters of A. M. Foley and R. C. King [Eur. J. Comb. 70, 325–353 (2018; Zbl 1384.05168)]. In this extension, the factorial powers are replaced with an arbitrary sequence of polynomials, as in Sergeev-Veselov’s generalised Schur functions and Okada’s generalised Schur P- and Q-functions. We also offer a similar generalisation for the rational Schur functions. We derive Littlewood-type identities for our generalisations. These identities allow us to give new (unflagged) Jacobi-Trudi identities for the Foley-King factorial characters and for rational versions of the factorial Schur functions. We also propose an extension of the original Macdonald’s ninth variation of Schur functions to the case of symplectic and orthogonal characters, which helps us prove Nägelsbach-Kostka identities.
MSC:
| 05E05 | Symmetric functions and generalizations |
| 05E10 | Combinatorial aspects of representation theory |
| 14N15 | Classical problems, Schubert calculus |
Keywords:
Schur functions; Littlewood-type identities; Jacobi-Trudi identities; Foley-King factorial charactersCitations:
Zbl 1384.05168References:
| [1] | A. Aggarwal, A. Borodin, L. Petrov, and M. Wheeler. Free fermion six vertex model: symmetric functions and random domino tilings. Selecta Math. (N.S.), 29(3):Paper No. 36, 138, 2023. · Zbl 1512.05390 |
| [2] | A. B. Balantekin and I. Bars. Representations of supergroups. J. Math. Phys., 22(8):1810-1818, 1981. · Zbl 0547.22014 |
| [3] | G. Benkart, C. L. Shader, and A. Ram. Tensor product representations for orthosym-plectic Lie superalgebras. J. Pure Appl. Algebra, 130(1):1-48, 1998. · Zbl 0932.17008 |
| [4] | A. Borodin and G. Olshanski. The boundary of the Gelfand-Tsetlin graph: a new approach. Adv. Math., 230(4-6):1738-1779, 2012. · Zbl 1245.05131 |
| [5] | C. J. Cummins and R. C. King. Composite Young diagrams, supercharacters of U(M/N ) and modification rules. J. Phys. A, 20(11):3121-3133, 1987. · Zbl 0645.17002 |
| [6] | A. M. Foley and R. C. King. Factorial characters of the classical Lie groups. European J. Combin., 70:325-353, 2018. · Zbl 1384.05168 |
| [7] | A. M. Foley and R. C. King. Factorial Q-functions and Tokuyama identities for classical Lie groups. European J. Combin., 73:89-113, 2018. · Zbl 1393.05300 |
| [8] | A. M. Foley and R. C. King. Determinantal and Pfaffian identities for ninth variation skew Schur functions and Q-functions. European J. Combin., 93:Paper No. 103271, 31 pp, 2021. · Zbl 1458.05258 |
| [9] | A. M. Foley and R. C. King. Factorial supersymmetric skew Schur functions and ninth variation determinantal identities. Ann. Comb., 25(2):229-253, 2021. · Zbl 1466.05219 |
| [10] | W. Fulton and J. Harris. Representation Theory: A First Course, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. · Zbl 0744.22001 |
| [11] | R. Howe. Remarks on classical invariant theory. Trans. Amer. Math. Soc., 313(2):539-570, 1989. · Zbl 0674.15021 |
| [12] | M. Itoh. Schur type functions associated with polynomial sequences of binomial type. Selecta Math. (N.S.), 14(2):247-274, 2009. · Zbl 1228.05291 |
| [13] | N. Jing and N. Rozhkovskaya. Vertex operators arising from Jacobi-Trudi identities. Comm. Math. Phys., 346(2):679-701, 2016. · Zbl 1404.17036 |
| [14] | K. Koike. On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters. Adv. Math., 74(1):57-86, 1989. · Zbl 0681.20030 |
| [15] | K. Koike and I. Terada. Young-diagrammatic methods for the representation theory of the classical groups of type B n , C n , D n . J. Algebra, 107(2):466-511, 1987. · Zbl 0622.20033 |
| [16] | D. E. Littlewood. On orthogonal and symplectic group characters. J. London. Math. Soc., 30:121-122, 1955. · Zbl 0066.28003 |
| [17] | D. E. Littlewood. The theory of group characters and matrix representations of groups. AMS Chelsea Publishing, Providence, RI, 2006. Reprint of the second (1950) edition. · Zbl 1090.20001 |
| [18] | I. G. Macdonald. Schur functions: theme and variations. In Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992), volume 498 of Publ. Inst. Rech. Math. Av., pages 5-39. Univ. Louis Pasteur, Strasbourg, 1992. · Zbl 0889.05073 |
| [19] | I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathemati-cal Monographs. The Clarendon Press, Oxford University Press, New York, second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications. · Zbl 0824.05059 |
| [20] | A. I. Molev. Comultiplication rules for the double Schur functions and Cauchy iden-tities. Electron. J. Combin., 16(1):#R13, 2009. · Zbl 1182.05128 |
| [21] | J. Nakagawa, M. Noumi, M. Shirakawa, and Y. Yamada. Tableau representation for Macdonald’s ninth variation of Schur functions. In Physics and combinatorics, 2000 (Nagoya), pages 180-195. World Sci. Publ., River Edge, NJ, 2001. · Zbl 0990.05134 |
| [22] | M. Nakasuji, O. Phuksuwan, and Y. Yamasaki. On Schur multiple zeta functions: a combinatoric generalization of multiple zeta functions. Adv. Math., 333:570-619, 2018. · Zbl 1414.11104 |
| [23] | S. Okada. A generalization of Schur’s P -and Q-functions. Sém. Lothar. Combin., 81:Art. B81k, 50 pp, 2020. · Zbl 1447.05229 |
| [24] | A. Okounkov and G. Olshanski. Shifted schur functions. St. Petersburg Math. J., 9(2):239-300, 1998. · Zbl 0894.05053 |
| [25] | G. Olshanski. Cauchy-Szegő kernels for Hardy spaces on simple Lie groups. J. Lie Theory, 5(2):241-273, 1995. · Zbl 0860.43008 |
| [26] | G. Olshanski. Interpolation Macdonald polynomials and Cauchy-type identities. J. Combin. Theory Ser. A, 162:65-117, 2019. · Zbl 1401.05301 |
| [27] | G. I. Ol’shanskii. Unitary representations of infinite-dimensional pairs (G, K) and the formalism of R. Howe. In Representation of Lie groups and related topics, volume 7 of Adv. Stud. Contemp. Math., pages 269-463. Gordon and Breach, New York, 1990. · Zbl 0724.22020 |
| [28] | A. Patel, H. Patel, and A. Stokke. Orthosymplectic Cauchy identities. Ann. Comb., 26(2):309-327, 2022. · Zbl 1492.05014 |
| [29] | E. M. Rains. BC n -symmetric polynomials. Transform. Groups, 10(1):63-132, 2005. · Zbl 1080.33018 |
| [30] | A. N. Sergeev and A. P. Veselov. Jacobi-Trudy formula for generalized Schur poly-nomials. Mosc. Math. J., 14(1):161-168, 2014. · Zbl 1297.05244 |
| [31] | R. P. Stanley. Enumerative combinatorics. Vol. 2. Cambridge University Press, Cambridge, 1999. · Zbl 0928.05001 |
| [32] | A. Stokke. An orthosymplectic Pieri rule. Electron. J. Combin., 25(3):#P3.37, 2018. · Zbl 1395.05187 |
| [33] | A. Stokke and T. Visentin. Lattice path constructions for orthosymplectic determi-nantal formulas. European J. Combin., 58:38-51, 2016. · Zbl 1343.05167 |
| [34] | S. Sundaram. Tableaux in the representation theory of the classical Lie groups. In Invariant theory and tableaux (Minneapolis, MN, 1988), volume 19 of IMA Vol. Math. Appl., pages 191-225. Springer, New York, 1990. · Zbl 0707.22004 |
| [35] | H. Weyl. The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J., 1939. · JFM 65.0058.02 |
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