The structure of the Universal Exponential Solution of the Yang-Baxter Equation

S. Fomin and A. Kirillov have shown that exponential solutions of the Yang-Baxter equation give rise to generalized Schubert polynomials and corresponding symmetric functions, and they provided several equivalent descriptions of the local stationary algebra ${\cal A}_0$ defined by this equation. Here we show that ${\cal A}_0$ is isomorphic to the graded associative algebra formally generated by the elements $a, C_0, C_1, C_2,\ldots$ satisfying the relations $[a,C_i]=C_{i+1}$ and $ [C_i,C_j]=0$. The rank of $ C_i$ is $i+1$. It will follow that the Hilbert series of ${\cal A}_0$ is ${1\over (1-t)^2}{1\over 1-t^2}{1\over 1-t^3}\cdots\ .$

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Published 1 January 1994