What is a concomitant (and other questions on D.E. Littlewood's "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" )? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:35:34Z http://mathoverflow.net/feeds/question/23301 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23301/what-is-a-concomitant-and-other-questions-on-d-e-littlewoods-products-and-ple What is a concomitant (and other questions on D.E. Littlewood's "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" )? Steven Sam 2010-05-03T00:16:58Z 2010-05-03T07:36:44Z <p>I'm trying to understand the paper "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" by D.E. Littlewood (<a href="http://math.mit.edu/~ssam/papers/littlewood.pdf" rel="nofollow">link</a>), but I'm having trouble overcoming the language that he uses (it's from 1958, which might be the problem?).</p> <p>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. </p> <p>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.</p> <p>So concrete questions:</p> <ol> <li>What does he mean by concomitant?</li> <li>Does anyone understand the passage on p.25 (and can you please explain to me)?</li> </ol> http://mathoverflow.net/questions/23301/what-is-a-concomitant-and-other-questions-on-d-e-littlewoods-products-and-ple/23307#23307 Answer by Mariano Suárez-Alvarez for What is a concomitant (and other questions on D.E. Littlewood's "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" )? Mariano Suárez-Alvarez 2010-05-03T02:03:39Z 2010-05-03T07:36:44Z <p>If a group $G$ acts on an affine variety $X$, and $W$ is a $G$-module, a <em>covariant</em> on $X$ with values in $W$ is a regular function $X\to W$ which is $G$-equivariant.</p> <p>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 <em>concomitant</em> 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.</p>