New S function series and non-compact Lie groups

Yang, Mei; Wybourne, B. G.

doi:10.1088/0305-4470/19/17/016

Cited by 14 publications

(7 citation statements)

References 13 publications

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“…Littlewood [31] gave a set of Schur function series, which allowed him to formulate various identities in an extremely compact notation. These identities have been extended by King, Dehuai and Wybourne [26] and later by Yang and Wybourne [49]; we follow the presentation of the latter. An S -function series is an infinite formal sum of Schur functions given via a generating function.…”

Section: Littlewood-king-wybourne Infinite Series Of Schur Functionsmentioning

confidence: 84%

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A Hopf laboratory for symmetric functions

Fauser¹,

Jarvis²

2004

J. Phys. A: Math. Gen.

105

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An analysis of symmetric function theory is given from the perspective of the underlying Hopf and bi-algebraic structures. These are presented explicitly in terms of standard symmetric function operations. Particular attention is focussed on Laplace pairing, Sweedler cohomology for 1-and 2-cochains, and twisted products (Rota cliffordizations) induced by branching operators in the symmetric function context. The latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type. A commentary on related issues in the combinatorial approach to quantum field theory is given.

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Section: Littlewood-king-wybourne Infinite Series Of Schur Functionsmentioning

confidence: 84%

“…For a Hopf algebraic approach to plethysm see [43]. The other series are then derived in a similar manner, see [49]. S -function series come in pairs which are mutually inverse and consecutively named.…”

Section: Littlewood-king-wybourne Infinite Series Of Schur Functionsmentioning

confidence: 99%

A Hopf laboratory for symmetric functions

Fauser¹,

Jarvis²

2004

J. Phys. A: Math. Gen.

105

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“…Littlewood [23] introduced a set of infinite Schur function series which, much later allowed King [15] to formulate various identities and branching rules in an extremely compact notation. These identities have been extended by King, Dehuai and Wybourne [16] and later by Yang and Wybourne [37]; we follow the presentation of the latter 1 . A Schur function series is an infinite series of Schur functions often given via a generating function defining a formal power series.…”

Section: Plethysms and Outer Exponentiationmentioning

confidence: 89%

“…Generaing functions and relations between some of these are displayed in table 2, following [37]. While we are following in this section the habit found in the literature to derive Schur function series from the L series, the forthcoming sections are based on the M series to avoid frequent usage of the clumsy notation L −1 .…”

Section: )mentioning

confidence: 99%

“…{1 2 }{1} = {1 3 } + {21} , but we postpone to investigate this complication until section 4. 5 Analogously to branching by skewing, formal multiplicative branchings, for example {λ} → ((λ · M )) , {λ} → ((λ · D)) , {λ} → ((λ · B)) , arise in the case of the branching of unitary representations of non-compact Lie groups [37,17,18] to representations of a maximal compact Lie subgroup. An analogous generalisation of multiplicative branching, for example {λ} → ((λ · {π}⊗M )) might equally be considered.…”

Section: "Coproducts" Of Schur Function Series: Combinatoricsmentioning

confidence: 99%

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New branching rules induced by plethysm

Fauser¹,

Jarvis²,

King³

et al. 2006

J. Phys. A: Math. Gen.

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We derive group branching laws for formal characters of subgroups Hπ of GL(n) leaving invariant an arbitrary tensor T π of Young symmetry type π where π is an integer partition. The branchings GL(n) ↓ GL(n − 1) , GL(n) ↓ O(n) and GL(2n) ↓ Sp(2n) fixing a vector vi , a symmetric tensor gij = gji and an antisymmetric tensor fij = −fji , respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function sπ ≡ {π} by the basic M series of complete symmetric functions and the L = M −1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras, and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains π -generalized Newell-Littlewood formulae, and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for H 1 3 , H21 , and H3 , showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine, and in some instances non reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2-cocycles. The algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for GL(n) formal characters. Modification rules are beyond the scope of the present paper, but are briefly discussed. applications of Kronecker products is provided by [13]. A necessary concomitant of these techniques is the automation provided by a symbolic computer package such as SCHUR c [36]. Finally, some of these techniques have been found to generalize to the representation theory of non-compact groups [19,17,18] In a recent paper [10] the role of symmetric functions in relation to group representation theory has been re-considered from the viewpoint of the underlying Hopf algebraic structure. This structure is in fact well known in the combinatorial literature [34,33,31]. In [10] the formalism of branching rules was aligned with certain endomorphisms on the algebra of symmetric functions, called branching operators, derived from 1-cochains, for which the multiplicative cohomology of Sweedler [32] provides a natural analytical setting and classification. Standard branchings from generic symmetric functions to symmetric functions of orthogonal or symplectic type (the classic Newell-Littlewood theorems for the group reduction from GL(n) ↓ O(n) or GL(n) ↓ Sp(n) ) were found to be derived from certain 2-cocycles (for which associativity is guaranteed).This result then prompts the more general question of classifying arbitrary, non-cohomologous, 2cocycles, and the nature of any associated character theory and of the algebraic or group structures which might be entrained therewith. While an ab initio approa...

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New Schur Function Series

Ishikawa

Wakayama

1998

Journal of Algebra

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This paper presents some new product identities for certain summations of Schur functions. These identities are generalizations of some famous identities known to Littlewood and appearing in Macdonald's book. We refer to these identities as the ''Littlewood-type formulas.'' In addition, analogues for summations of characters of the other classical groups are also given. The Littlewood-type formulas in this paper are separated into two classes, the rational Schur function series and the generalized Schur function series. An application of a rational Schur function series to the infinite product representation of the elliptic theta functions is also given. We prove these Littlewood-type formulas using the Cauchy᎐Binet formula. The Cauchy᎐Binet formula is a basic but powerful tool applicable in the present context, which can be derived from our Pfaffian formula, as we explain.

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New S function series and non-compact Lie groups

Cited by 14 publications

References 13 publications

A Hopf laboratory for symmetric functions

A Hopf laboratory for symmetric functions

New branching rules induced by plethysm

New Schur Function Series

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