# Cut-and-join equation and Schur function identity

This is somewhat related to my last MO post:

sum of the character of the symmetric group

Let $p_n$ be the $n$-th Newton symmetric function, and $s_{\nu}$ be the Schur function indexed by the partition $\nu$. We know that $$\sum_{\nu} s_{\nu}(x) s_{\nu}(y) = \prod_{i,j} \frac{1}{1-x_i y_j}$$ where the summation is over all partitions, including the empty one.

We also have $$\sum_{\nu} s_{\nu}(x)= \frac{1}{\prod_i(1-x_i)\prod_{i>j}(1-x_i x_j)}$$ see Macdonald's Symmetric functions and Hall polynomials, page 76. These two formulas are both of "infinite sum = infinite product" type.

In my current research, I was encountered with the sum $H(t) := \sum_{\nu} e^{\frac{\kappa_{\nu}}{2} t}s_{\nu}(x)$, where $\kappa_{\nu} = \sum_{i=1}^{l(\nu)} \nu_i(\nu_i-2i+1)$. Note that $H(t)$ satisfies the cut-and-join equation $$\frac{\partial}{\partial t} H(t) = \Delta H(t)$$ with initial value $H(0) = \sum_{\nu} s_{\nu}$. Here $$\Delta := \frac{1}{2}\sum_{i,j}((i+j)p_i p_j \frac{\partial}{\partial p_{i+j}} + ijp_{i+j}\frac{\partial}{\partial p_{i}}\frac{\partial}{\partial p_{j}})$$ is the cut-and-join operator. This is because Schur functions are the eigenfunctions of $\Delta$, namely, $$\Delta s_{\nu} = \frac{\kappa_{\nu}}{2} s_{\nu}$$.

My question is: Is there an infinite product expression for $H(t)$?

Any comment is welcome!

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I am not sure if there is any good infinite product expression for H(t). But I think one can obtain at least some integral transform of such infinite product. Namely, the operator $\exp (t \Delta)$ can be represented as integral of the shift operator (as in the reference I gave you), which acts on the infinite product H(0) preserving its structure. – Sasha Jan 9 '12 at 8:30