# Commutators of Schur polynomials of Lie algebra elements

Question:

Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, is there a nice way to work this out?

If it helps, the algebra in question is easily treatable as the $x_i$ and $\frac{\partial}{\partial x_i}$ acting on the left on $\mathbb{C}[x_1, x_2, \ldots]$, which is usually the context the Schur polynomials appear in...I have Schur polynomials $f_n$ and $g_n$ in the variables $\frac{\mu}{k} x_k$ and $-\frac{\mu}{k} \frac{\partial}{\partial x_k}$ respectively for some parameter $\mu$, and want to know about $$[f_m, g_n]$$ in general, and $$[f_n, g_n]$$ in particular.

Context:

I have an operator $V_\mu(z)$ mapping between `bosonic Fock spaces' $\mathcal{F}_\eta$, the countable-dimensional vector spaces cyclically generated from highest weight vectors $\left| \eta \right>$ by acting freely with the Heisenberg algebra $\mathcal{H} = \mathrm{span}_\mathbb{C}\{\mathbf{1}, \alpha_n: \; n \in \mathbb{Z}\}$. $$\mathcal{F}_\eta = \mathcal{U}(\mathcal{H}) \cdot \left| \eta \right>$$ where the positive-indexed operators annihilate the highest weight vector; $$\alpha_n \cdot \left| \eta \right> = 0 \quad \forall n > 0 \\ \alpha_0 \cdot \left| \eta \right> = \eta \left| \eta \right>\\ \mathbf{1} \cdot \left| \eta \right> = \left| \eta \right>$$ and $\mathcal{H}$ has the following well-known commutation relations: $$[\alpha_m, \alpha_n] = m \delta_{m, -n} \\ [\mathbf{1}, \alpha_n] = 0 \\$$

$V_\mu(z): \mathcal{F}_\eta \to \mathcal{F}_{\eta + \mu}$ is a screening operator, i.e. $$V_\mu(z) = e^{\mu q} z^{\mu \eta} \exp\left( \mu \sum_{n > 0}\frac{\alpha_{-n}}{n}z^n \right) \exp\left(-\mu\sum_{n>0}\frac{\alpha_n}{n}z^{-n}\right)$$ which as you can see has exponentials of sums of independent elements, the generating function of the Schur polynomials. We get $$V_\mu(z) = e^{\mu q} z^{\mu \eta}\sum_{m \in \mathbb{Z}}\sum_{n \in \mathbb{Z}} f_m g_n z^{m-n}$$ where $f_n$ and $g_n$ are the $n$th elementary Schur polynomial in variables $\frac{\mu}{k} \alpha_{-k}$ and $-\frac{\mu}{k}\alpha_k$ respectively.

I am interested in the behaviour of the Fourier modes of this function, taking $$V_\mu(z) = \sum_{n \in \mathbb{Z}} V_\mu^{(n)} z^{-n-1}$$ and in particular whether they have nice commutation relations.

• Applying a polynomial to an element of a Lie algebra isn't a priori a meaningful operation. You're applying polynomials to elements of an algebra (not just Lie algebra) of differential operators. – Qiaochu Yuan Jan 16 '15 at 8:54
• Yes, that's true. The operation is that of left multiplication in the universal enveloping algebra of the Lie algebra. – Michael Cromer Jan 16 '15 at 8:58
• That is almost but not quite the case. The Weyl algebra is almost but not quite a universal enveloping algebra; it's the quotient of a universal enveloping algebra by the additional relation that a particular central element acts as the identity. – Qiaochu Yuan Jan 16 '15 at 16:38
• Crossposted: mathoverflow.net/questions/194069/… – Qiaochu Yuan Jan 16 '15 at 16:51