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Background

When constructing the exterior algebra of a (finite-dimensional, complex) vector space $V$, there are two equivalent pictures. The first is the quotient picture. First you define the tensor algebra $T(V)$, define $\mathcal{J}$ to be the 2-sided ideal generated by elements of the form $x\otimes y + y \otimes x$, and then define the exterior algebra to be the quotient $\Lambda(V) = T(V)/\mathcal{J}$.

The other viewpoint is via embedding the exterior algebra in the tensor algebra. This is done as follows. There is an action $\rho$ of the symmetric group $S_n$ on $V^{\otimes n}$ for each $n$, given by $$\rho_\pi (v_1 \otimes \dots \otimes v_n) = v_{\pi(1)} \otimes \dots \otimes v_{\pi(n)}.$$ Then the map $$A_n = \frac{1}{n!} \sum_{\pi \in S_n}sgn(\pi) \rho_\pi$$ is idempotent, i.e. satisfies $A_n^2 = A_n$. If we define $$ A = \bigoplus_{n=0}^\infty A_n$$ on $T(V)$, then it turns out that the kernel of $A$ is equal to $\mathcal{J}$, so that $\Lambda(V) \simeq \mathrm{im}(A)$.

Why I care

I'm trying to understand the quantum analogue of this, where $V$ is the fundamental representation/vector representation of $U_q(\mathfrak{sl}_N)$. The problem is that there is no longer an action of the symmetric group on $V^{\otimes n}$; instead there is an action of the braid group. Both the quotient picture and the embedding picture have analogues in the quantum setting. In particular, denote by $\sigma$ the braiding on $V \otimes V$, and let $\sigma_i$ be the automorphism of $V^{\otimes n}$ given by $\sigma$ acting in the $i$ and $i+1$ spots.

One constructs the $q$-antisymmetrizer as follows. Define $\tau_i$ to be the adjacent transpositions $(i, i+1)$ in $S_n$. For a permutation $\pi$, write $\pi= \tau_{i_1} \dots \tau_{i_k},$ where $k$ is the minimal number of adjacent transpositions needed. Then define $$\sigma_\pi = \sigma_{i_1} \dots \sigma_{i_k}.$$ It is a theorem that this is well-defined, i.e. that any two minimal decompositions of $\pi$ as products of adjacent transpositions can be transformed into one another using only the braid relations. Anyway, once this is known, you can define the $q$-analogue of the antisymmetrizer map as $$A_n = \frac{q^{\binom{n}{2}}}{[n]!} \sum_{\pi \in S_n} sgn(\pi) q^{-\ell(\pi)} \sigma_\pi,$$ where $[n]$ is the $q$-number and $\ell(\pi)$ is the length of a minimal decomposition of $\pi$. The significance of the $\binom{n}{2}$ is that it is the length of the longest word in $S_n$. As far as I can tell, this was first defined by Jimbo in his 1986 paper "A $q$-analogue of $U(\mathfrak{gl}_{N+1})$, Hecke Algebra, and the Yang-Baxter Equation". He states that $A_n^2 = A_n$, and refers to Gyoja's paper "A $q$-Analogue of Young Symmetrizer." Gyoja's paper certainly does the trick, but I find that something is lost in the abstraction.

I've verified that $A_n^2 = A_n$ for $n = 3$, which already involves 36 terms, and it wasn't that enlightening. The problem is that the braid generators $\sigma_i$ aren't idempotent, they instead satisfy $\sigma_i^2 = 1 + (q - q^{-1})\sigma_i$.

The question

Does anybody know a nice way of seeing that the $q$-antisymmetrizer is idempotent, or a nice presentation of it somewhere?

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2 Answers

up vote 6 down vote accepted

This element has many expressions. It is characterised up to a scalar multiple by the property that $\sigma_iA_n=-q^{-1}A_n$ for $i=1,2,\ldots ,n-1$. It also satisfies $A_n\sigma_i=-q^{-1}A_n$ for $i=1,2,\ldots ,n-1$. In particular it is central.

Using this property you can calculate $A_n^2$. Note that $\sigma_\pi A_n=(-q)^{-\ell(\pi)}= sgn(\pi)q^{-\ell(\pi)}$. So $$A_n.\sum_\pi sgn(\pi)q^{-\ell(\pi)}\sigma_\pi=A_n.\sum_\pi q^{-2\ell(\pi)}$$

You fix the scalar factor by the condition $A_n^2=A_n$. Alternatively in the one dimensional representation $\sigma_i \mapsto -q^{-1}$ you require $A_n \mapsto 1$.

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That's a nice argument. I guess that's more or less the same as what you do for the Young symmetrizers as well, right? –  MTS Jul 29 '10 at 8:24
    
Sure. Just change $-q^{-1}$ to $q$. P.S. I have fixed the signs. –  Bruce Westbury Jul 29 '10 at 8:51
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Group Theory: Birdtracks, Lie's, and Exceptional Groups by Predrag Cvitanovic has a graphical way of presenting the idempotents. His book is available online here:

http://birdtracks.eu/

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