Groups generated by involutions acting on combinatorial objects

In this expository talk I will discuss various groups generated by involutions acting on objects familiar to algebraic combinatorics. I will survey: relations in the group; the isomorphism type of the group, viewed as a permutation group; and special elements which behave well, especially because of their algebraic significance. A prototypical example is the group generated by Bender-Knuth involutions acting on semistandard or standard Young tableaux. This example was thoroughly investigated by Berenstein and Kirillov, among others. Another example is the "toggle group" acting on the set of order ideals of a finite poset, which was first studied by Cameron and Fon-der-Flaass. By recasting these two examples in terms of piecewise-linear maps, I will explain that there is in fact a precise connection between them.