# What is a concomitant (and other questions on D.E. Littlewood's “Products and plethysms of characters with orthogonal, symplectic, and symmetric groups” )?

I'm trying to understand the paper "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" by D.E. Littlewood (link), but I'm having trouble overcoming the language that he uses (it's from 1958, which might be the problem?).

In particular, one word he uses a lot is "concomitant". A google search for the definition turned out to be extremely unhelpful, but I think this is something really basic that a lot of people here know.

As a bonus, I'm really just trying to understand the passage on p.25 between Theorem VII and Theorem VIII. As I understand it, the "fundamental forms" he mentions are the functions $S_i$ defined on p.24, but when applying these to $\bigwedge^2 {\bf C}^n$, I'm getting 0, but $\bigwedge^2 {\bf C}^n$ this is not an irreducible representation of the symmetric group $\mathfrak{S}_n$ (he seems to be claiming that the intersection of the kernels of the fundamental forms should be). The reference he mentions doesn't seem to be of much help either.

So concrete questions:

1. What does he mean by concomitant?
2. Does anyone understand the passage on p.25 (and can you please explain to me)?
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I think I understand the answers to the questions above now. The key was to read D.E. Littlewood, Invariant theory, tensors and group characters, Philos. Trans. Roy. Soc. London. Ser. A. 239, (1944). 305--365 in case anyone else was interested. – Steven Sam Aug 12 '10 at 2:11

If a group $G$ acts on an affine variety $X$, and $W$ is a $G$-module, a covariant on $X$ with values in $W$ is a regular function $X\to W$ which is $G$-equivariant.
In the special case in which $G=\mathrm{SL}(V)$ is the special linear group on a vector space $V$, $X=\mathrm{Pol}_{d_1}(V)\otimes\cdots\otimes\mathrm{Pol}_{d_s}(V)$ and $W=\mathrm{Pol}_{d}(V)$, with natural actions of $G$, a covariant $X\to W$ is called a concomitant of degree $d$. The canonical example of a concomitant is the resultant $R(f_1,\dots,f_s)$ of $s$ homogeneous polynomial functions $f_1\in\mathrm{Pol}_{d_1}(V), \dots, f_s\in\mathrm{Pol}_{d_s}(V)$ of degrees $d_1,\dots,d_s$, which has degree $0$. A simpler example is the Jacobian of $n$ homogeneous forms in $n$ variables.
These modern definitions are probably not what Littlewood had in mind. The relevant definitions can be found in references on classical invariant theory like Grace and Young or Glenn. In the case $X\rightarrow W$ above a classic would call that a covariant rather than a concomitant. The latter would be obtained when W involves both Pol_d(V) and Pol_d(V*) tensors factors. – Abdelmalek Abdesselam Jul 17 '10 at 22:04