Two semidirect products

· Monomial matrices Definition

The normalizer consists of invertible monomial matrices. Choose the permutation representative \(w\) by \(we_i=e_{w(i)}\), and define

$$\unif^{\lam}:=\operatorname{diag}(\unif^{\lam_1},\ldots,\unif^{\lam_n}),\qquad\lam\in\ZZ^n.$$

Modulo diagonal units every normalizer class has a unique representative \(\unif^{\lam}w\).

· Multiply the two data Proved here

$$w\muv=(\muv_{w^{-1}(1)},\ldots,\muv_{w^{-1}(n)}),$$ $$(\unif^{\lam}w)(\unif^{\muv}v)=\unif^{\lam+w\muv}wv.$$

Proof

1. Conjugation by a permutation matrix gives \(w\unif^{\muv}w^{-1}=\unif^{w\muv}\).
2. Insert \(w^{-1}w\) between the diagonal and permutation factors and collect them.

· Affine versus extended Definition

$$\Qco=\{\lam\in\ZZ^n:\textstyle\sum_i\lam_i=0\},\qquad\Wa=\Qco\rtimes S_n,$$ $$\kap:\Wt\longrightarrow\ZZ,\qquad\kap(\unif^{\lam}w)=\textstyle\sum_i\lam_i=\operatorname{val}\det(\unif^{\lam}w),\qquad\ker\kap=\Wa.$$

For \(n=1\), \(\Wa\) is trivial; for \(n\ge2\), it is the affine Coxeter group of type \(\widetilde A_{n-1}\).

· Length and the alcove stabilizer Definition

The length counts affine root hyperplanes separating the base alcove and its image; for Schubert cells it equals their dimension.

$$\ell(w)=\dim(\Iwa w\Iwa/\Iwa),\qquad\Om=\{w\in\Wt:\ell(w)=0\}.$$

Length zero means preserving the base alcove as a set. In GLₙ the central scalar translations are invisible on the reduced apartment, so this stabilizer is infinite.

$$\ell(\unif^{\lam})=\sum_{1\le i<j\le n}|\lam_i-\lam_j|.$$

The translation formula counts the parallel hyperplanes \(x_i-x_j\in\ZZ\) crossed in each root direction; the central direction contributes none.