Plethysms, replicated Schur functions and series, with applications to vertex operators
Fauser, B and Jarvis, PD and King, RC, Plethysms, replicated Schur functions and series, with applications to vertex operators, Journal of Physics A:Mathematical and Theoretical , 43, (40) pp. 1-30. ISSN 1751-8113 (2010) [Refereed Article]
Specializations of Schur functions are exploited to define and evaluate the
Schur functions sλ[αX] and plethysms sλ[αsν(X))] for any α—integer, real or
complex. Plethysms are then used to define pairs of mutually inverse infinite
series of Schur functions, Mπ and Lπ , specified by arbitrary partitions π.
These are used in turn to define and provide generating functions for formal
λ , of certain groups Hπ , thereby extending known results for
orthogonal and symplectic group characters. Each of these formal characters
is then given a vertex operator realization, first in terms of the seriesM = M(0)
and various L⊥σ dual to Lσ , and then more explicitly in the exponential form.
Finally the replicated form of such vertex operators are written down.
The characters of the orthogonal and symplectic groups have been
found by Schur  and Weyl  respectively. The method used is
transcendental, and depends on integration over the group manifold.
These characters, however, may be obtained by purely algebraic
methods, . . . . This algebraic method would seem to offer a better prospect
of successful application to other restricted groups than the method of
Littlewood D E 1944 Phil. Trans. R. Soc. London, Ser. A 239 (809)
PACS numbers: 02.10.−v, 02.10.De, 02.20.−a, 02.20.Hj
Mathematics Subject Classification: 05E05, 17B69, 11E57, 16W30,
20E22, 33D52, 43A40