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Let $S^\pm$ be the half-spin representations of $\mathfrak{so}_{2n}\mathbb{C}$. Fulton-Harris's Representation Theory says on page 378 that the character $D^\pm$ of $S^\pm$ is the sum $$\sum x_1^{\pm 1/2}\cdot...\cdot x_n^{\pm 1/2},$$ "where the number of signs is even or odd according to the sign." This notation is confusion to me- if for example $n=3$ and we're working with $\mathfrak{so}_6\mathbb{C}$, then is $D^+$ the third elementary symmetric polynomial in $x_1^{1/2}+x_1^{-1/2}, x_2^{1/2}+x_2^{-1/2}, x_3^{1/2}+x_3^{-1/2}$?

Explicitly I think my question boils down to: could you write out the terms in the characters $D^\pm$ of the half-spin representations?

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I think you should read that chapter more carefully. I do not have the book at hand, but it should be clear. – Bombyx mori May 4 '13 at 5:28

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