Cox rings of moduli of quasi parabolic principal bundles and the K-Pieri rule

We study a toric degeneration of the Cox ring of the moduli of principal $SL_m(\mathbb{C})$ bundles on the projective line, with quasi parabolic data given by the the stabilizer of the highest weight vector in $\mathbb{C}^m$ and its dual $\bigwedge^{m-1}(\mathbb{C}^m)$. The affine semigroup algebra resulting from this degeneration is described using the $K-$Pieri rule from Kac-Moody representation theory. Along the way we give a proof of the $K-$Pieri rule which utilizes the classical Pieri rule and elements of commutative algebra, and we describe a relationship between the Cox ring and a classical invariant ring studied by Weyl.Comment: 26 pages, 11 figures, final versio

Pre-Darwinian species change: reincarnation and transformism in George Sand’s Évenor et Leucippe

No abstract available

Generalized Laguerre Unitary Ensembles and an interacting particles model with a wall

We introduce and study a new interacting particles model with a wall and two kinds of interactions - blocking and pushing - which maintain particles in a certain order. We show that it involves a random matrix model.Comment:

Orbit measures, random matrix theory and interlaced determinantal processes

A connection between representation of compact groups and some invariant ensembles of Hermitian matrices is described. We focus on two types of invariant ensembles which extend the Gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction mulltiplicities. We show that a large class of them are determinantal

Compactifications of character varieties and skein relations on conformal blocks

Let $M_C(G)$ be the moduli space of semistable principal $G-$bundles over a smooth curve $C$. We show that a flat degeneration of this space $M_{C_{\Gamma}}(G)$ associated to a singular stable curve $C_{\Gamma}$ contains the free group character variety $\mathcal{X}(F_g, G)$ as a dense, open subset, where $g = genus(C).$ In the case $G = SL_2(\mathbb{C})$ we describe the resulting compactification explicitly, and in turn we conclude that the coordinate ring of $M_{C_{\Gamma}}(SL_2(\mathbb{C}))$ is presented by homogeneous skein relations. Along the way, we prove the parabolic version of these results over stable, marked curves $(C_{\Gamma}, \vec{p}_{\Gamma})$.Comment: 36 page, 24 figures, final versio

Interacting particle models and the Pieri-type formulas : the symplectic case with non equal weights

We have introduced recently a particles model with blocking and pushing interactions which is related to a Pieri type formula for the orthogonal group. This model has a symplectic version presented here