Let $Symm$ be the vector space with basis $(b\_\lambda)$ given by the
set of all partitions $\lambda$ (of all natural numbers), thought
of as Young diagrams. Let $e\_i$ be
the *degree $i$ Pieri operator* that takes 
$b\_\lambda \mapsto \sum b\_{\lambda'}$ where $\lambda'$ has $i$ more
boxes than $\lambda$, no two in the same column. The $(e\_i)$ commute,
because they correspond to multiplication by Schur functions in
the ring of symmetric functions.


With respect to this basis, we can talk about $e\_i^T$,
which takes $b\_{\lambda'} \mapsto \sum b\_{\lambda}$ where again,
$\lambda'$ has $i$ more boxes than $\lambda$, no two in the same column.


1. What are the commutation relations between the $(e\_i)$ and the $(e\_i^T)$?
2. I presume that the algebra of operators on $Symm$ so generated
is the Heisenberg algebra (on infinitely many generators) acting on its Fock space. How does one write the differentiation 
operator $\frac{d}{db\_i}$ in terms of the $(e\_i,e\_i^T)$?
3. What is a good reference for this presentation of the Heisenberg algebra?



Heisenberg algebra from Pieri operators and their transposes?
I believe that you mean for your basis $b\_\lambda$ to correspond to the basis of Schur functions in the ring of symmetric functions, in which case your $e\_i$ operator corresponds to multiplication with respect to the complete symmetric function $h\_i = b\_{(i)} = b\_i$ and your $e\_i^T$ corresponds to the adjoint of multiplication by $h\_i$ with respect to the Hall inner product.


Supposing this is the case, we can view the ring of symmetric functions as an algebra of operators acting on itself where we view elements in the ring of symmetric functions, $\Lambda = \mathbb{Z}[p\_1, p\_2, \dots]$, as polynomials in the power sums, $p\_i$, and the action of $p\_i$ on an element in $\Lambda$ is by multiplication.


In this case the adjoint of multiplication by $p\_i$ with respect to the Hall inner product is $p\_i^\perp = i \frac{d}{d p\_i}$. When viewed in this way, the algebra generated by the $p\_i$ and $p\_i^\perp$ is the Heisenberg or oscillator algebra.


As mentioned above in S. Carnahan's answer, this can be found in Kac's *Infinite Dimensional Lie Algebras* and Kac and Raina's *Bombay Lectures*. This can also be found, from a symmetric function perspective, in Macdonald's book *Symmetric Functions and Hall Polynomials* Chapter I, Section 5, Exercise 3 which begins on page 75 (Here I am referring to the second edition).


I'm not sure what the commutation relations are between $h\_i$ and $h\_i^\perp$ (I am abusing notation here and using $h\_i$ to mean multiplication by $h\_i$), however, I can verify the suspicion stated in your response to Ben Webster's answer.


It is shown in Macdonald that for any function $f \in \Lambda$, $f(p\_1, p\_2, \dots)^\perp = f(p\_1^\perp, p\_2^\perp, \dots)$ thus $h\_1^\perp = p\_1^\perp = \frac{d}{dp\_1}$ and so is a first order linear differential operator. In particular, chain rule works nicely and we see that $$ h\_1^\perp \circ h\_i = h\_i \circ h\_1^\perp + h\_1^\perp(h\_i) = h\_i \circ h\_1^\perp + h\_{i-1} $$ where I have used $\circ$ to denote composition. Also note that $h\_1^\perp(h\_i) = h\_{i-1}$ follows from the description of $h\_1^\perp = e\_1^T$ and the fact that $h\_i = b\_i = b\_{(i)}$ in the notation of the OP.


I'm also not sure about how to express $\frac{d}{dh\_i}$ in terms of $h\_i$ and $h\_i^\perp$, however, there may be some threads in Macdonald's book.


As for finding a nice reference for all of this, I haven't been able to find one. What I was able to piece together from the references given earlier essentially became my [Master's Thesis](http://uwspace.uwaterloo.ca/handle/10012/4770) although the purpose was somewhat different. In particular we were looking at the Bernstein operator which is related to all of this in that it arises via the Boson-Fermion Correspondence and is related to Integrable Hierarchies. It also arises in Macdonald's book I 5 Ex 29 on page 95. We took a very combinatorial approach, which can be found [here](http://arxiv.org/abs/0902.4441). A more algebraic approach was taken by Hird, Jing and Stitzinger ([here](http://arxiv.org/abs/1010.4072)) and in particular, their approach extends to the B-type case (Schur Q Functions).


