3,528 research outputs found
Schur times Schubert via the Fomin-Kirillov algebra
We study multiplication of any Schubert polynomial by a
Schur polynomial (the Schubert polynomial of a Grassmannian
permutation) and the expansion of this product in the ring of Schubert
polynomials. We derive explicit nonnegative combinatorial expressions for the
expansion coefficients for certain special partitions , including
hooks and the 2x2 box. We also prove combinatorially the existence of such
nonnegative expansion when the Young diagram of is a hook plus a box
at the (2,2) corner. We achieve this by evaluating Schubert polynomials at the
Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the
nonnegativity conjecture of Fomin and Kirillov.
This approach works in the more general setup of the (small) quantum
cohomology ring of the complex flag manifold and the corresponding (3-point)
Gromov-Witten invariants. We provide an algebro-combinatorial proof of the
nonnegativity of the Gromov-Witten invariants in these cases, and present
combinatorial expressions for these coefficients
Inversion of some series of free quasi-symmetric functions
We give a combinatorial formula for the inverses of the alternating sums of
free quasi-symmetric functions of the form F_{\omega(I)} where I runs over
compositions with parts in a prescribed set C. This proves in particular three
special cases (no restriction, even parts, and all parts equal to 2) which were
conjectured by B. C. V. Ung in [Proc. FPSAC'98, Toronto].Comment: 6 page
Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
We revisit the work of the first named author and using simpler algebraic
arguments we calculate integrals of polynomial functions with respect to the
Haar measure on the unitary group U(d). The previous result provided exact
formulas only for 2d bigger than the degree of the integrated polynomial and we
show that these formulas remain valid for all values of d. Also, we consider
the integrals of polynomial functions on the orthogonal group O(d) and the
symplectic group Sp(d). We obtain an exact character expansion and the
asymptotic behavior for large d. Thus we can show the asymptotic freeness of
Haar-distributed orthogonal and symplectic random matrices, as well as the
convergence of integrals of the Itzykson-Zuber type
On the diagram of 132-avoiding permutations
The diagram of a 132-avoiding permutation can easily be characterized: it is
simply the diagram of a partition. Based on this fact, we present a new
bijection between 132-avoiding and 321-avoiding permutations. We will show that
this bijection translates the correspondences between these permutations and
Dyck paths given by Krattenthaler and by Billey-Jockusch-Stanley, respectively,
to each other. Moreover, the diagram approach yields simple proofs for some
enumerative results concerning forbidden patterns in 132-avoiding permutations.Comment: 20 pages; additional reference is adde
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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Asymptotics of characters of symmetric groups, genus expansion and free probability
The convolution of indicators of two conjugacy classes on the symmetric group
S_q is usually a complicated linear combination of indicators of many conjugacy
classes. Similarly, a product of the moments of the Jucys--Murphy element
involves many conjugacy classes with complicated coefficients. In this article
we consider a combinatorial setup which allows us to manipulate such products
easily: to each conjugacy class we associate a two-dimensional surface and the
asymptotic properties of the conjugacy class depend only on the genus of the
resulting surface. This construction closely resembles the genus expansion from
the random matrix theory. As the main application we study irreducible
representations of symmetric groups S_q for large q. We find the asymptotic
behavior of characters when the corresponding Young diagram rescaled by a
factor q^{-1/2} converge to a prescribed shape. The character formula (known as
the Kerov polynomial) can be viewed as a power series, the terms of which
correspond to two-dimensional surfaces with prescribed genus and we compute
explicitly the first two terms, thus we prove a conjecture of Biane.Comment: version 2: change of title; the section on Gaussian fluctuations was
moved to a subsequent paper [Piotr Sniady: "Gaussian fluctuations of
characters of symmetric groups and of Young diagrams" math.CO/0501112
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