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Title: Combinatorics of Gaudin systems : cactus groups and the RSK algorithm
Author: White, Noah Alexander Matthias
ISNI:       0000 0004 6420 9380
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2016
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This thesis explores connections between the Gaudin Hamiltonians in type A and the combinatorics of tableaux. The cactus group acts on standard tableaux via the Schützenberger involution. We show in this thesis that the action of the cactus group on standard tableaux can be recovered as a monodromy action of the cactus group on the simultaneous spectrum of the Gaudin Hamiltonians. More precisely, we consider the action of the Bethe algebra, which contains the Gaudin Hamiltonians, on the multiplicity space of a tensor product of irreducible glr-modules. The spectrum of this algebra forms a flat and finite family over M0,n+1(C). We use work of Mukhin, Tarasov and Varchenko, who link this spectrum to certain Schubert intersections, and work of Speyer, who extends these Schubert intersections to a flat and finite map over the entire moduli space of stable curves M0,n+1(C). We show the monodromy over the real points M0,n+1(R) can be identified with the action of the cactus group on a tensor product of irreducible glr-crystals. Furthermore we show this identification is canonical with respect to natural labelling sets on both sides.
Supervisor: Wemyss, Michael ; Gordon, Iain Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: representation theory ; general linear group ; combinatorics ; Gaudin spin chain model ; Bethe ansatz ; Schützenberger involution ; Gaudin Hamiltonians