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By Borovik A. V.

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2, we have a right from now on to identify n and n with the corresponding subalgebras of n . Then n is a free right n -module on ∈ n . As another consequence, if m ≤ n, we can consider m a basis x as the subalgebra of n generated by x1 xm , s1 sm−1 . 2: ∈ n+ as a basis. 1. 3 The center of n The following simple description of the center is very important. 1 The center of x1 xn . 9). Conversely, take a central element z = w∈Sn fw w ∈ n where each fw ∈ n . Let w be maximal with respect to the Bruhat order such that n with wi = i.

Moreover, each vector wT is a simultaneous eigenvector for Ln−k+1 Ln ∈ n−k k with eigenvalues res Tk , respectively. 6 Let / be a skew shape with / k 1 2 = −1 L / = k. Then / if / is a skew hook, otherwise. 0 The following is a very effective way to evaluate an irreducible character on a given element. 7 (Murnaghan–Nakayama rule) Let / be a skew shape with / = k, and c be an element of Sk whose cycle shape corresponds to a partition = 1 ≥ · · · ≥ l > 0 ∈ k . Then / c = −1 LH H where the sum is over all sequences H of partitions = 0 ⊂ 1 ⊂ ··· ⊂ l = such that i / i − 1 is a skew hook with 1 ≤ i ≤ l, and L H = li=1 L i / i − 1 .

3 Degenerate affine Hecke algebra In this chapter we define the degenerate affine Hecke algebra n . As a vector xn of the group algebra space, n is the tensor product FSn ⊗ F x1 xn . Moreover, FSn and the free commutative polynomial algebra F x1 xn are subalgebras of n isomorphic to FSn and FSn ⊗ 1 and 1 ⊗ F x1 xn , respectively. Furthermore, there exists an algebra homomorF x1 phism n → FSn , which is the “identity” on the subalgebra FSn , that is sends w ⊗1 to w, see Chapter 7. 1. In particular, the center of n is what we would like it to be: the ring of symmetric polynomials xn Sn .

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