
%  When you are satisfied with your abstract, please delete everything
%  up to this point, and also the closing line (\end{document}).

%  Put the relevent information between the braces below:

\titleofpresentation{Standard tableaux and Klyachko's Theorem on Lie representations}

\presenter{Marianne Johnson}  %  Name just one speaker, please; collaborators
            %  can be credited in the body of the abstract.

%  [Body of abstract goes here.]
Let $n \in \mathbb{N}$ and let $\lambda = (\lambda_{1}, \ldots,
\lambda_{k})$ be a partition of $n$. That is, $\lambda_{1} \geq
\cdots \geq \lambda_{k} > 0$ and $\lambda_{1} + \cdots +\lambda_{k}
= n$. The Young diagram corresponding to $\lambda$ is a collection
of $n$ boxes arranged in left justified rows with $\lambda_{i}$
boxes in the $i$th row. A standard tableau of shape $\lambda$ is a
numbering of the Young diagram of $\lambda$ with the numbers from
$\{1, \ldots, n\}$ in such a way that the entries strictly increase
along every row and down every column. We say that an entry $i$ is a
descent in a standard tableau $T$ if the entry $i+1$ occurs in any
row below the row in which $i$ appears. The sum of all descents in
$T$ is called the major index of $T$.

For example, $\lambda =(5,3,2,1)$ is a partition of 11. Shown below
is a standard tableau $T$ of shape $\lambda$. The descents of $T$
are $2, 6, 9$ and $10$. Hence, the major index of $T$ is $2+6+9+10 =
27$.

\begin{picture}(125,125)
\put(25,100){\line(1,0){100}} \put(25,80){\line(1,0){100}}
\put(25,60){\line(1,0){60}} \put(25,40){\line(1,0){40}}
\put(25,20){\line(1,0){20}} \put(25,100){\line(0,-1){80}}
\put(45,100){\line(0,-1){80}} \put(65,100){\line(0,-1){60}}
\put(85,100){\line(0,-1){40}} \put(105,100){\line(0,-1){20}}
\put(125,100){\line(0,-1){20}}

\put(30,85){\large{1}} \put(50,85){\large{2}} \put(70,85){\large{5}}
\put(90,85){\large{6}} \put(110,85){\large{9}}
\put(30,65){\large{3}} \put(50,65){\large{4}}
\put(68,65){\large{10}} \put(30,45){\large{7}}
\put(50,45){\large{8}} \put(28,25){\large{11}}
\end{picture}

We show that for all partitions $\lambda$ of $n \geq 3$ ($\lambda
\neq (2^{2}), (2^{3}), (1^{n}), (n)$) there exists a standard
tableau of shape $\lambda$ with major index coprime to $n$
\cite{johnson1}. By a result of Kra\'{s}kiewicz and Weyman
\cite{johnson3}, this provides a new purely combinatorial proof of
Klyachko's Theorem \cite{johnson2} on Lie representations of the
general linear group.

\begin{ourbiblio}{99}  %  Delete this environment if you
               %  aren't giving a bibliography.

\bibitem{johnson1} Marianne Johnson. Standard tableaux and Klyachko's Theorem on
Lie representations. \textit{Journal of Combinatorial Theory, Series
A},{\bf 114} (2007), no. 1, 151--158.

\bibitem{johnson2}A.A. Klyachko. Lie elements in the tensor algebra.
\textit{Sibirsk Mat. {\v Z}}, {\bf 15} (1974) 1296–-1304 (Russian).
English translation: \textit{Siberian J. Math.} {\bf 15} (1974),
914--921.

\bibitem{johnson3} Witold Kra\'{s}kiewicz and Jerzy Weyman. Algebra of coinvariants
and the action of a Coxeter element. \textit{Bayreuth. Math. Schr.},
{\bf 63} (2001), 265–-284. (Preprint, 1987).

\end{ourbiblio}

\presenteraddress{University of Manchester}  % The name of a university is fine.

\email{marianne.johnson@maths.man.ac.uk}

\web{personalpages.manchester.ac.uk/postgrad/M.Johnson-3/}
% Those without homepages should delete this.

% Delete the next line when you are done.
