40 research outputs found
Stanley character polynomials
Stanley considered suitably normalized characters of the symmetric groups on
Young diagrams having a special geometric form, namely multirectangular Young
diagrams. He proved that the character is a polynomial in the lengths of the
sides of the rectangles forming the Young diagram and he conjectured an
explicit form of this polynomial. This Stanley character polynomial and this
way of parametrizing the set of Young diagrams turned out to be a powerful tool
for several problems of the dual combinatorics of the characters of the
symmetric groups and asymptotic representation theory, in particular to Kerov
polynomials.Comment: Dedicated to Richard P. Stanley on the occasion of his seventieth
birthda
Partial transpose of random quantum states: exact formulas and meanders
We investigate the asymptotic behavior of the empirical eigenvalues
distribution of the partial transpose of a random quantum state. The limiting
distribution was previously investigated via Wishart random matrices indirectly
(by approximating the matrix of trace 1 by the Wishart matrix of random trace)
and shown to be the semicircular distribution or the free difference of two
free Poisson distributions, depending on how dimensions of the concerned spaces
grow. Our use of Wishart matrices gives exact combinatorial formulas for the
moments of the partial transpose of the random state. We find three natural
asymptotic regimes in terms of geodesics on the permutation groups. Two of them
correspond to the above two cases; the third one turns out to be a new matrix
model for the meander polynomials. Moreover, we prove the convergence to the
semicircular distribution together with its extreme eigenvalues under weaker
assumptions, and show large deviation bound for the latter.Comment: v2: change of title, change of some methods of proof
Linear versus spin: representation theory of the symmetric groups
We relate the linear asymptotic representation theory of the symmetric groups
to its spin counterpart. In particular, we give explicit formulas which express
the normalized irreducible spin characters evaluated on a strict partition
with analogous normalized linear characters evaluated on the double
partition . We also relate some natural filtration on the usual
(linear) Kerov-Olshanski algebra of polynomial functions on the set of Young
diagrams with its spin counterpart. Finally, we give a spin counterpart to
Stanley formula for the characters of the symmetric groups.Comment: 41 pages. Version 2: new text about non-oriented (but orientable)
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Polynomial functions on Young diagrams arising from bipartite graphs
We study the class of functions on the set of (generalized) Young diagrams
arising as the number of embeddings of bipartite graphs. We give a criterion
for checking when such a function is a polynomial function on Young diagrams
(in the sense of Kerov and Olshanski) in terms of combinatorial properties of
the corresponding bipartite graphs. Our method involves development of a
differential calculus of functions on the set of generalized Young diagrams.Comment: To appear in DMTCS pro
Zonal polynomials via Stanley's coordinates and free cumulants
We study zonal characters which are defined as suitably normalized
coefficients in the expansion of zonal polynomials in terms of power-sum
symmetric functions. We show that the zonal characters, just like the
characters of the symmetric groups, admit a nice combinatorial description in
terms of Stanley's multirectangular coordinates of Young diagrams. We also
study the analogue of Kerov polynomials, namely we express the zonal characters
as polynomials in free cumulants and we give an explicit combinatorial
interpretation of their coefficients. In this way, we prove two recent
conjectures of Lassalle for Jack polynomials in the special case of zonal
polynomials.Comment: 45 pages, second version, important change
Skew Howe duality and random rectangular Young tableaux
We consider the decomposition into irreducible components of the external
power regarded as a
-module. Skew Howe duality
implies that the Young diagrams from each pair which
contributes to this decomposition turn out to be conjugate to each other,
i.e.~. We show that the Young diagram which corresponds
to a randomly selected irreducible component has the same
distribution as the Young diagram which consists of the boxes with entries
of a random Young tableau of rectangular shape with rows and
columns. This observation allows treatment of the asymptotic version of this
decomposition in the limit as tend to infinity.Comment: 17 pages. Version 2: change of title, section on bijective proofs
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