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?


4 Answers 4


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 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. A more algebraic approach was taken by Hird, Jing and Stitzinger (here) and in particular, their approach extends to the B-type case (Schur Q Functions).

  • $\begingroup$ Nice answer (it was almost the one I wanted to give). The only thing I would add is that the De Concini, Kac, Kazhdan paper "Boson-Fermion Correspondence over $\mathbb{Z}$" describes a generating series for the $h_i^{\perp}$. $\endgroup$
    – David Hill
    May 18, 2011 at 18:51

There is a presentation in section 14.10 of Kac's Infinite dimensional Lie algebras, or in Kac and Raina's Bombay Lectures, where your $e_i$ and $e_i^T$ are called $\alpha_i$ and $\alpha_{-i}$. I believe your space of partitions is isomorphic to the charge one subspace of the Fock space of semi-infinite wedges by the following rule: turn your Young diagram diagonally so its vertex is pointing down at $x$-coordinate one, and glue it to the graph of $y=|x-1|$. If you normalize the edge lengths to be $\sqrt{2}$, you get integer $x$-coordinates of corners. The upper envelope of the resulting figure is an infinite collection of line segments that slope upward or downward. The $x$-coordinates of the bottoms of the downward-sloping edges give you a basis vector of the Fock space.

I don't see a definition of $b_i$ in your question, so I don't know how to differentiate with respect to it.

  • $\begingroup$ I think that the graphical construction of the partition that Scott is talking about goes by the name of Maya diagrams. Googling that will no doubt yield concrete references where the OPs question might be answered. $\endgroup$ May 18, 2011 at 8:43
  • $\begingroup$ Thank you, José. I had not heard that name. I quite like Peter Tingley's brief (15 page) notes: www-math.mit.edu/~ptingley/lecturenotes/Fock_space-2010.pdf $\endgroup$
    – S. Carnahan
    May 18, 2011 at 9:13
  • $\begingroup$ You're welcome, Scott. I only heard that name recently, after a good part of a lifetime of exposure to semi-infinite forms! I wonder who introduced this name. $\endgroup$ May 18, 2011 at 9:35
  • $\begingroup$ $b_i = b_\lambda$ where $\lambda$ has one part, $i$. $Symm$ is a polynomial ring in the $(b_i)$. $\endgroup$ May 18, 2011 at 11:49
  • $\begingroup$ $\S$14.10 doesn't seem to involve these. Can you tell me where to look in Kac-Raina? $\endgroup$ May 18, 2011 at 12:29

I learned of this construction from Leclerc and Thibon, though that paper is a pretty considerable generalization to the quantum case, and incorporating $U_q(\mathfrak{\widehat{sl}}_n)$-actions. In that paper, you should take $n=1$ and your $e_i$ is called $V_i$ and $e_i^T$ is called $U_i$. Presumably there are also older references (certainly Leclerc and Thibon speak about it as though it were well-known). You might try the book of MacDonald, but I can't preview it online, so I don't know either way.

  • $\begingroup$ (1) is definitely false; $e_1^T e_2$ on the empty partition is $b_1$, whereas $e_2 e_1^T$ is zero. I suspect $[e_1^T,e_i] = e_{i-1}$. The $e_i$ in Leclerc-Thibon cares which diagonal things are added in, unless $n=1$ (the ${\mathfrak gl}(1)$ case, which I guess is what I want), but then only gets my $e_1$. $\endgroup$ May 18, 2011 at 12:34
  • $\begingroup$ Right, that's what I meant by a pretty considerable generalization. That doesn't change the fact that they do have your operators when n=1. $\endgroup$
    – Ben Webster
    May 18, 2011 at 16:01
  • $\begingroup$ I've deleted the relations Allen was referring to, since he was right that it was wrong. $\endgroup$
    – Ben Webster
    May 18, 2011 at 16:02

The relation between symmetric functions and infinite dimensional Lie algebras is usually attributed to Sato and Sato, but it might be that the original paper(s) is(are) only in Japanese.

An accessible exposition and references in

Solitons and Infinite Dimensional Lie Algebras By Michio JIMBO and Tetsuji MIWA

Publ RIMS, Kyoto Univ. 19 (1983), 943-1001


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