Annales de l’Institut Henri Poincaré D

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Volume 3, Issue 2, 2016, pp. 163–255
DOI: 10.4171/AIHPD/28

Published online: 2016-05-19

Type $\widehat{\mathrm C}$ Brauer loop schemes and loop model with boundaries

Anita Ponsaing[1] and Paul Zinn-Justin[2]

(1) Universite Pierre et Marie Curie, Paris, France
(2) Université Pierre et Marie Curie, Paris, France

In this paper we study the Brauer loop model on a strip and the associated quantum Knizhnik–Zamolodchikov (qKZ) equation. We show that the minimal degree solution of the Brauer qKZ equation with one of four dierent possible boundary conditions, gives the multidegrees of the irreducible components of generalizations of the Brauer loop scheme of [16, Knutson–Zinn-Justin ’07] with one of four kinds of symplectic-type symmetry. This is accomplished by studying these irreducible components, which are indexed by link patterns, and describing the geometric action of Brauer generators on them. We also provide recurrence relations for the multidegrees and compute the sum rules (multidegrees of the whole schemes).

Keywords: Brauer algebra, quantum Knizhnik–Zamolodchikov equation, equivariant cohomology, Loop model

Ponsaing Anita, Zinn-Justin Paul: Type $\widehat{\mathrm C}$ Brauer loop schemes and loop model with boundaries. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 3 (2016), 163-255. doi: 10.4171/AIHPD/28