By Andrew Pressley

To take inventory and to debate the main fruitful instructions for destiny examine, a number of the world's major figures met on the Durham Symposium on Quantum teams in the summertime of 1999, and this quantity offers a good evaluate of the cloth provided there. It contains vital surveys of either cyclotomic Hecke algebras and the dynamical Yang-Baxter equation. Plus contributions which deal with the development and type of quantum teams or the linked recommendations of the quantum Yang-Baxter equation. The illustration idea of quantum teams is mentioned, as is the functionality algebra method of quantum teams, and there's a new examine the origins of quantum teams within the idea of integrable structures.

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1 . 2 . 2 D 3 . 3 . 3 . 4 . 4 5 . 5 . 6 B 7 . 7 . 8 9 m . . . . 3 4 5 6 7 8 9 10 . . . . . 1 . 2 . 2 A 3 . 4 6 7. C 8 10 E m Figure 13: Conﬁguration of all 4 rectangles with central line. We consider the left and right hand corner points in this conﬁguration.

4 . 4 5 . 5 . 6 . 6 7 . 7 . 8 ... ... ...... ... ... ... . .. ... ... ... ... . . ... . . ... . ... . ... ... . ... ... . ..... ... ... ... ... . ... . ... ... ... ... ... ... ... . . ... ... ... ... ... ..... .. 5 . 6 7 . 7 . 8 9 − LL − − − − − − A (0, 7, 2, 9)-rectangle . . . 4 5 6 7 8 9 4 5 6 7 8 9 10 . 6 . 7 . 8 . − − − RRR − − − A (3, 7, 7, 11)-rectangle ..... ...... ... ... ... ... . . ... . ... . ... ... . ... . ...

N + 1 lie in S but i − 1 ∈ S. Then edge i − 1 of P is labelled R and is the leftmost labelled edge in P . (d) The elements of S not in an initial segment as in (b) or a terminal segment as in (c) give rise to edges L or R in P as in (a). Example. Let n = 13. If S = {1, 2, 3, 4, 7, 8, 11}, then P = − − LRRLLRR − − −. See Figure 6 for an explanatory diagram. Applying this bijection S → P to the chamber sets obtained from a reduced word i we obtain a set of 12 n(n − 1) partial quivers associated with i.