Irreducible Highest Weight Vectors

Let $G$ be an affine, connected, reductive group and $X$ a $G$-module. Choose a maximal torus $T\subseteq G$, a Borel $B\subseteq G$ containing $T$ and let $U$ be the unipotent radical of $B$. Denote by the character group of $T$. Let $\Lambda\subseteq\mathbb{X}$ be the set of dominant weights of $G$ with respect to these choices. We can decompose the graded coordinate ring $\mathbb{C}[X]=\bigoplus_{\lambda\in\Lambda} V_{(\lambda)}$ into its isotypic components $V_{(\lambda)}$ of weight $\lambda$. Let $\Lambda_X=\{ \lambda\in\Lambda \mid V_{(\lambda)}\ne\{0\}\}$ be the set of weights that occur in $\mathbb{C}[X]$. Let $V_{(\lambda)}\cong V_\lambda^{\oplus n_\lambda}$, where $V_\lambda$ is the irreducible module of highest weight $\lambda$. Each $V_\lambda$ has a highest weight vector which is unique up to scaling - let $f_{\lambda 1}, \ldots, f_{\lambda n_\lambda}\in V_{(\lambda)}$ be linearly independent highest weight vectors. Claim. *For each $1\le k\le n_\lambda$, if the function $f:=f_{\lambda k}\in\mathbb{C}[X]$ is reducible, then there exist weights $\lambda_1,\ldots,\lambda_r\in\Lambda_X$ such that $\lambda$ is an $\mathbb N$-linear combination of the $\lambda_i$.* Indeed, about a year ago this statement was completely unclear to me. However, it's actually not that hard to see and I felt like sharing my proof. (more…)

A determinant for reductive groups

For the group $G=\mathrm{GL}_n(\mathbb{K})$ of invertible matrices over any field, there is the well-known determinant character $\det:G\to\mathbb{K}^\times$ which, well, maps any matrix to its determinant. It has the property that $[G,G]=\mathrm{SL}_n(\mathbb{K})=\{ g\in G \mid \det(g)=1 \},$ where $[G,G]$ denotes the derived subgroup of $G$. In more geometric terms, $[G,G]$ is the vanishing of the regular function $\det-1$ on $G$. Because I find it rather cute, I will show that you can find a similar function for all linear, reductive groups. (more…)

Characters of the Symmetric Group

Harm Derksen brachte uns schon im Jahre 1998 die weltschnellste Methode zur Berechnung irreduzibler Charaktere der symmetrischen Gruppe. Aber in seinem preprint, Computing with Characters of the Symmetric Group, bleiben einige Fragen offen, die ich hiermit zu klären versuche.
» The code to end all codes «

Die Welt ist einen halben Herzschlag alt. Der unsterbliche Prophet meditiert seit Anbeginn der Laufzeit auf dem তাজমহল und studiert die heiligen 12 Zeilen C-Code, die jede SAT Instanz in $\mathcal{O}(n^{9699690})$ lösen. Wenn der Puls eine Ganzzahl wird, so wird er erwachen und die Wahrheit verkünden, und Lob wird gepriesen, und Licht wird ewig scheinen auf die Kinder des Schaltkreises. Denn er ist der Taktgeber, und an der Spitze des Signals wird es sein, wenn er uns erscheint. Buch Arithmæl Circulæ, Vers 21:7 Wollen Sie mehr wissen?