Cauchy-Littlewood identities

We follow Chapter 1 of [{macdonald98}].

Definition

$\sum_\lambda s_\lambda(x) s_\lambda(y) = \prod{1 \over 1 - x_i y_j} =: \Pi(x, y) \qquad (4.3)$

Unless otherwise specified, sums like this are over all partitions. This is true for symmetric polynomial case $$x = x_{1 : n}$$ and $$y = y_{1 : \ell}$$ as well as in the symmetric function case.

recall the definition

$z_\lambda = \prod_{i \ge 1} i^{m_i} m_i!$

where $$m_i$$ is the number of parts of $$\lambda$$ equal to $$i$$, i.e. $$m_i = \lambda'_i - \lambda'_{i + 1}$$.

Other similar stuff:

\begin{align} \sum_\lambda z_\lambda^{-1} p_\lambda(x) p_\lambda(y) &= \Pi(x, y) \qquad(4.1) \\ \sum_\lambda h_\lambda(x) m_\lambda(y) = \sum_\lambda h_\lambda(y) m_\lambda(x) &= \Pi(x, y) \qquad(4.2) \end{align}

Claim. $$\omega_x \Pi(x, y) = \omega_y \Pi(x, y) = \prod(1 + x_i y_j) =: \Pi'(x, y)$$.

Proof.

\begin{align} \omega_x \Pi(x, y) = \omega_x \prod_j H(y_j, x) = \omega_x \prod_j (\sum_n y_j^n h_n(x)) = \prod_j (\sum_n y_j^n \omega_x h_n(x))\\ = \prod_j \sum_n y_j^n e_n(x) = \prod_j E(y_j, x) = RHS. \end{align}

$$\square$$

Orthogonality

Define scalar product on $$\Lambda$$ by

$\langle h_\lambda, m_\mu \rangle = \delta_{\lambda \mu}$

Then

Theorem (4.6). For $$n \ge 0$$, let $$(u_\lambda)_{\lambda \vdash n}$$ and $$(v_\lambda)_{\lambda \vdash n}$$ be $$\mathbb Q$$ bases of $$\Lambda^n_{\mathcal Q}$$. Then the following are equivalent:

• $$\langle u_\lambda, v_\mu \rangle = \delta_{\lambda \mu}, \forall \lambda, \mu$$
• $$\sum_\lambda u_\lambda(x) v_\lambda(y) = \Pi(x, y)$$

Hence

$\langle p_\lambda, p_\mu \rangle = \delta_{\lambda \mu} z_{\lambda}, \qquad \langle s_\lambda, s_\mu \rangle = \delta_{\lambda \mu}.$

(4.9) the bilinear form is symmetric and positive definite.

Since $$\omega(p_\lambda) = \pm p_\lambda$$,

(4.10) The involution $$\omega$$ is an isometry, i.e. $$\langle \omega u, \omega v \rangle = \langle u, v \rangle$$.

Therefore by the Claim we have three dual Cauchy-Littlewood identities:

\begin{align} \sum_\lambda \epsilon_\lambda z_\lambda^{-1} p_\lambda(x) p_\lambda(y) &= \Pi'(x, y) \\ \sum_\lambda m_\lambda(x) e_\lambda(y) = \sum_\lambda m_\lambda(y) e_\lambda(x) &= \Pi'(x, y) \\ \sum_\lambda s_\lambda(x) s_{\lambda'}(y) &= \Pi'(x, y). \end{align}

References

• [macdonald98] Symmetric Functions and Hall Polynomials, , 1998.