arXiv subjects

Guillaume Pouchin

Publications and source records attributed to Guillaume Pouchin.

5 recordsLinked to original sources

arXiv2016

Representation theory and an isomorphism theorem for the Framisation of the Temperley-Lieb algebra

In this paper, we describe the irreducible representations and give a dimension formula for the Framisation of the Temperley-Lieb algebra. We then prove that the Framisation of the Temperley-Lieb algebra is isomorphic to a direct sum of matrix algebras over tensor products of classical Temperley-Lieb algebras. This allows us to construct a basis for it. We also study in a similar way the Complex Reflection Temperley-Lieb algebra.

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arXiv2015

Higgs bundles on weighted projective lines and loop crystals

We consider the space of nilpotent Higgs bundles on a weighted projective line, as a global analog of the nilpotent cone. We show that it is pure, compute its dimension, and define geometric correspondences between irreducible components. We prove it by constructing a loop analog of a cristal, in the spirit of Kashiwara and Saito, for some corresponding loop Kac-Moody algebra.

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arXiv2014

Determination of the representations and a basis for the Yokonuma-Temperley-Lieb algebra

We determine the representations of the Yokonuma-Temperley-Lieb algebra, which is defined as a quotient of the Yokonuma-Hecke algebra by generalising the construction of the classical Temperley-Lieb algebra. We then deduce the dimension of this algebra, and produce an explicit basis for it.

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arXiv2010

Higgs algebra of curves and loop crystals

We define the Higgs algebra $\mathcal{H}_\P1$ of the projective line, as a convolution algebra of constructible functions on the global nilpotent cone $\underlineΛ_\P1$, a lagrangian substack of the Higgs bundle $T^*\Coh_\P1$, where $\Coh_\P1$ is the stack of coherent sheaves on $\P1$. We prove that $\mathcal{H}_\P1$ is isomorphic to (some completion of) $U^+(\hat{sl}_2)$. We use this geometric realization to define a semicanonical basis of $U^+(\hat{sl}_2)$, indexed by irreducible components of $\underlineΛ_\P1$. We also construct a combinatorial data on this set of irreducible components in the spirit of \cite{KS}, which is an affine analog of a crystal. We call it a loop crystal and give some of its properties.

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arXiv2008

A geometric Schur-Weyl duality for quotients of affine Hecke algebras

After establishing a geometric Schur-Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts have nice ideals coming from geometry, allowing duality for quotients. Some of the quotients of the positive affine Hecke algebra are then identified to some cyclotomic Hecke algebras and the geometric setting allows the construction of canonical bases.

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