267 research outputs found
A two-dimensional pictorial presentation of Berele's insertion algorithm for symplectic tableaux
We give the first two-dimensional pictorial presentation of Berele's
correspondence \cite{Berele}, an analogue of the Robinson-Schensted (R-S)
correspondence \cite{Robinson, Schensted} for the symplectic group Sp(2n, \Cpx
). From the standpoint of representation theory, the R-S correspondence
combinatorially describes the irreducible decomposition of the tensor powers of
the natural representation of GL(n,\Cpx). Berele's insertion algorithm gives
the bijection that describes the irreducible decomposition of the tensor powers
of the natural representation of Sp(2n, \Cpx). Two-dimensional pictorial
presentations of the R-S correspondence via local rules (first given by S.
Fomin \cite{Fomin,FominGen}) and its many variants have proven very useful in
understanding their properties and creating new generalizations. We hope our
new presentation will be similarly useful.Comment: 42 page
Algebraic aspects of increasing subsequences
We present a number of results relating partial Cauchy-Littlewood sums,
integrals over the compact classical groups, and increasing subsequences of
permutations. These include: integral formulae for the distribution of the
longest increasing subsequence of a random involution with constrained number
of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as
new proofs of old formulae; relations of these expressions to orthogonal
polynomials on the unit circle; and explicit bases for invariant spaces of the
classical groups, together with appropriate generalizations of the
straightening algorithm.Comment: LaTeX+amsmath+eepic; 52 pages. Expanded introduction, new references,
other minor change
Random walks and random fixed-point free involutions
A bijection is given between fixed point free involutions of
with maximum decreasing subsequence size and two classes of vicious
(non-intersecting) random walker configurations confined to the half line
lattice points . In one class of walker configurations the maximum
displacement of the right most walker is . Because the scaled distribution
of the maximum decreasing subsequence size is known to be in the soft edge GOE
(random real symmetric matrices) universality class, the same holds true for
the scaled distribution of the maximum displacement of the right most walker.Comment: 10 page
Descent sets for symplectic groups
The descent set of an oscillating (or up-down) tableau is introduced. This
descent set plays the same role in the representation theory of the symplectic
groups as the descent set of a standard tableau plays in the representation
theory of the general linear groups. In particular, we show that the descent
set is preserved by Sundaram's correspondence. This gives a direct
combinatorial interpretation of the branching rules for the defining
representations of the symplectic groups; equivalently, for the Frobenius
character of the action of a symmetric group on an isotypic subspace in a
tensor power of the defining representation of a symplectic group.Comment: 22 pages, 2 figure
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