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Factorial characters of the classical Lie groups

Factorial characters of the classical Lie groups
Factorial characters of the classical Lie groups

Just as the definition of factorial Schur functions as a ratio of determinants allows one to show that they satisfy a Jacobi–Trudi-type identity and have an explicit combinatorial realisation in terms of semistandard tableaux, so we offer here definitions of factorial irreducible characters of the classical Lie groups as ratios of determinants that share these two features. These factorial characters are each specified by a partition, λ=(λ12,…,λn), and in each case a flagged Jacobi–Trudi identity is derived that expresses the factorial character as a determinant of corresponding factorial characters specified by one-part partitions, (m), for which we supply generating functions. These identities are established by manipulating determinants through the use of certain recurrence relations derived from these generating functions. The transitions to combinatorial realisations of the factorial characters in terms of tableaux are then established by means of non-intersecting lattice path models. The results apply to gl(n), so(2n+1), sp(2n) and o(2n), and are extended to the case of so(2n) by making use of newly defined factorial difference characters.

0195-6698
325-353
Foley, Angèle M.
53e0c35d-e4bb-461d-9638-1d1c5fdc6f27
King, Ronald C.
76ae9fb3-6b19-449d-8583-dbf1d7ed2706
Foley, Angèle M.
53e0c35d-e4bb-461d-9638-1d1c5fdc6f27
King, Ronald C.
76ae9fb3-6b19-449d-8583-dbf1d7ed2706

Foley, Angèle M. and King, Ronald C. (2018) Factorial characters of the classical Lie groups. European Journal of Combinatorics, 70, 325-353. (doi:10.1016/j.ejc.2018.01.011).

Record type: Article

Abstract

Just as the definition of factorial Schur functions as a ratio of determinants allows one to show that they satisfy a Jacobi–Trudi-type identity and have an explicit combinatorial realisation in terms of semistandard tableaux, so we offer here definitions of factorial irreducible characters of the classical Lie groups as ratios of determinants that share these two features. These factorial characters are each specified by a partition, λ=(λ12,…,λn), and in each case a flagged Jacobi–Trudi identity is derived that expresses the factorial character as a determinant of corresponding factorial characters specified by one-part partitions, (m), for which we supply generating functions. These identities are established by manipulating determinants through the use of certain recurrence relations derived from these generating functions. The transitions to combinatorial realisations of the factorial characters in terms of tableaux are then established by means of non-intersecting lattice path models. The results apply to gl(n), so(2n+1), sp(2n) and o(2n), and are extended to the case of so(2n) by making use of newly defined factorial difference characters.

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More information

Accepted/In Press date: 24 January 2018
e-pub ahead of print date: 18 February 2018
Published date: 1 May 2018

Identifiers

Local EPrints ID: 419432
URI: https://eprints.soton.ac.uk/id/eprint/419432
ISSN: 0195-6698
PURE UUID: 24309b48-1faa-4f5e-be3e-e66067c78577

Catalogue record

Date deposited: 12 Apr 2018 16:30
Last modified: 13 Mar 2019 18:38

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