Reflection and algorithm proofs of some more lie group dual pair identities

Recently developed reflection techniques of Gessel and Zeilberger and Schensted algorithm techniques of Benkart and Stroomer are used to give new proofs of some dual pair (or Cauchy-type) symmetric function identifies first found by A. O. Morris long ago and recently found anew by Hasegawa in the context of dual pairs of representations of Lie algebras

Random Walk in an Alcove of an Affine Weyl Group, and Non-Colliding Random Walks on an Interval

We use a reflection argument, introduced by Gessel and Zeilberger, to count the number of k-step walks between two points which stay within a chamber of a Weyl group. We apply this technique to walks in the alcoves of the classical affine Weyl groups. In all cases, we get determinant formulas for the number of k-step walks. One important example is the region m>x_1>x_2>...>x_n>0, which is a rescaled alcove of the affine Weyl group C_n. If each coordinate is considered to be an independent particle, this models n non-colliding random walks on the interval (0,m). Another case models n non-colliding random walks on the circle.Comment: v.2, 22 pages; correction in a definition led to changes in many formulas, also added more background, references, and example

Brownian Motion in a Weyl Chamber, Non-Colliding Particles, and Random Matrices

Let $n$ particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is $A_{n-1}$, the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time $t$ without having collided by time $t$. We show that the probability that there will be no collision up to time $t$ is asymptotic to a constant multiple of $t^{-n(n-1)/4}$ as $t$ goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group $B_n$ gives a model of $n$ independent particles with a wall at $x=0$. We can define Brownian motion on a Lie algebra, viewing it as a vector space; the eigenvalues of a point in the Lie algebra correspond to a point in the Weyl chamber, giving a Brownian motion conditioned never to exit the chamber. If there are $m$ roots in $n$ dimensions, this shows that the radial part of the conditioned process is the same as the $n+2m$-dimensional Bessel process. The conditioned process also gives physical models, generalizing Dyson's model for $A_{n-1}$ corresponding to ${\mathfrak s}{\mathfrak u}_n$ of $n$ particles moving in a diffusion with a repelling force between two particles proportional to the inverse of the distance between them

Skew Howe duality and limit shapes of Young diagrams

We consider the skew Howe duality for the action of certain dual pairs of Lie groups $(G_1, G_2)$ on the exterior algebra $\bigwedge(\mathbb{C}^{n} \otimes \mathbb{C}^{k})$ as a probability measure on Young diagrams by the decomposition into the sum of irreducible representations. We prove a combinatorial version of this skew Howe for the pairs $(\mathrm{GL}_{n}, \mathrm{GL}_{k})$, $(\mathrm{SO}_{2n+1}, \mathrm{Pin}_{2k})$, $(\mathrm{Sp}_{2n}, \mathrm{Sp}_{2k})$, and $(\mathrm{Or}_{2n}, \mathrm{SO}_{k})$ using crystal bases, which allows us to interpret the skew Howe duality as a natural consequence of lattice paths on lozenge tilings of certain partial hexagonal domains. The $G_1$-representation multiplicity is given as a determinant formula using the Lindstr\"om-Gessel-Viennot lemma and as a product formula using Dodgson condensation. These admit natural $q$-analogs that we show equals the $q$-dimension of a $G_2$-representation (up to an overall factor of $q$), giving a refined version of the combinatorial skew Howe duality. Using these product formulas (at $q =1$), we take the infinite rank limit and prove the diagrams converge uniformly to the limit shape.Comment: 54 pages, 12 figures, 2 tables; v2 fixed typos in Theorem 4.10, 4.14, shorter proof of Theorem 4.6 (thanks to C. Krattenthaler), proved of Conjecture 4.17 in v

Asymptotics for random walks in alcoves of affine Weyl groups

Asymptotic results are derived for the number of random walks in alcoves of affine Weyl groups (which are certain regions in $n$-dimensional Euclidean space bounded by hyperplanes), thus solving problems posed by Grabiner [J. Combin. Theory Ser. A 97 (2002), 285-306]. These results include asymptotic expressions for the number of vicious walkers on a circle, and as well for the number of vicious walkers in an interval. The proofs depart from the exact results of Grabiner [loc. cit.], and require as diverse means as results from symmetric function theory and the saddle point method, among others.Comment: 72 pages, AmS-LaTeX; major revision: there are now also theorems on the asymptotic enumeration with non-fixed end points in types B and D; a flaw in the statement and proof of Lemma A has been correcte