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Let $p_1, p_2, \ldots$, be the power sum symmetric functions. Let $p_n^* = n \frac{\partial}{\partial p_n}$. Then $$ p_n^* p_m - p_m p_n^* = \delta_{m, n} 1. $$ Where could I find this result in some books or papers? Thank you very much.

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This result follows trivially via the chain rule and from the fact that the power sum symmetric functions are algebraically independent. That the star represents adjoint with respect to the Hall inner product can be found in Macdonald's book Symmetric Functions and Hall Polynomials Chapter I, Section 5, Exercise 3 which begins on page 75 in the Second Edition. –  Sean Carrell Jun 20 '13 at 11:45
    
@Sean, thank you very much. –  Jianrong Li Jun 21 '13 at 3:21
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up vote 2 down vote accepted

In Nakajima's annals paper on the homlogy of Hilbert Schemes and the representation theorx of the Heisenberg algebra.

This is also in his book on Hilbert schemes of points in chapter 8.

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Here is a PhD thesis on representations of the infinite dim. Heisenberg group.

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@Peter, thank you very much. But it seems that this result is not proved in the thesis. I have to show that the power sum symmetric functions satisfies the equation: $[p_n^*, p_m]=\delta_{m,n}1$. –  Jianrong Li Jun 20 '13 at 6:27
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