Abstract
The up-operators $u_i$ and down-operators $d_i$ (introduced as Schur operators by Fomin) act on partitions by adding/removing a box to/from the $i$th column if possible. It is well known that the $u_i$ alone satisfy the relations of the (local) plactic monoid, and the present authors recently showed that relations of degree at most 4 suffice to describe all relations between the up-operators. Here we characterize the algebra generated by the up- and down-operators together, showing that it can be presented using only quadratic relations.
Highlights
The up-operators ui for i ∈ N act on a partition λ by adding a box to the ith column of λ if the result is a partition and by sending λ to 0 otherwise
The down-operators di act on λ by subtracting a box from the ith column if the result is a partition and by sending it to 0 otherwise. These operators were introduced as Schur operators by Fomin [1] and further discussed by Fomin and Greene [2] in the context of noncommutative Schur functions
For |i − j| 2, (In particular, the ui satisfy the classical Knuth relations of the plactic monoid—see for instance [4].) The current authors proved in [5] that the ui satisfy the additional degree 4 relation ui+1ui+2ui+1ui = ui+1ui+2uiui+1 and that this relation along with the local plactic relations characterize the algebra generated by the ui, therein called the algebra of Schur operators
Summary
The up-operators ui for i ∈ N act on a partition λ by adding a box to the ith column of λ if the result is a partition and by sending λ to 0 otherwise. The down-operators di act on λ by subtracting a box from the ith column if the result is a partition and by sending it to 0 otherwise These operators were introduced as Schur operators by Fomin [1] and further discussed by Fomin and Greene [2] in the context of noncommutative Schur functions. (In particular, the ui satisfy the classical Knuth relations of the plactic monoid—see for instance [4].) The current authors proved in [5] (see Meinel [6]) that the ui satisfy the additional degree 4 relation ui+1ui+2ui+1ui = ui+1ui+2uiui+1 and that this relation along with the local plactic relations characterize the algebra generated by the ui, therein called the algebra of Schur operators It was noted in [1] (using the fact that the down-operators can be thought of as transposes of the up-operators) that the di satisfy: djdi = didj didi+1di = dididi+1, di+1didi+1 = didi+1di+1, for |i − j| 2, and that together the ui and di satisfy: diuj = ujdi d1u1 = id, di+1ui+1 = uidi.
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