108 research outputs found
Comultiplication rules for the double Schur functions and Cauchy identities
The double Schur functions form a distinguished basis of the ring
\Lambda(x||a) which is a multiparameter generalization of the ring of symmetric
functions \Lambda(x). The canonical comultiplication on \Lambda(x) is extended
to \Lambda(x||a) in a natural way so that the double power sums symmetric
functions are primitive elements. We calculate the dual Littlewood-Richardson
coefficients in two different ways thus providing comultiplication rules for
the double Schur functions. We also prove multiparameter analogues of the
Cauchy identity. A new family of Schur type functions plays the role of a dual
object in the identities. We describe some properties of these dual Schur
functions including a combinatorial presentation and an expansion formula in
terms of the ordinary Schur functions. The dual Littlewood-Richardson
coefficients provide a multiplication rule for the dual Schur functions.Comment: 44 pages, some corrections are made in sections 2.3 and 5.
A new quantum analog of the Brauer algebra
We introduce a new algebra B_l(z,q) depending on two nonzero complex
parameters such that B_l(q^n,q) at q=1 coincides with the Brauer algebra
B_l(n). We establish an analog of the Brauer-Schur-Weyl duality where the
action of the new algebra commutes with the representation of the twisted
deformation U'_q(o_n) of the enveloping algebra U(o_n) in the tensor power of
the vector representation.Comment: 13 page
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