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BN-pair (Definition)

Let $ G$ be a group. Then $ G$ has a $ BN$-pair or a Tits system if the following conditions hold:

  1. $ B$ and $ N$ are subgroups of $ G$ such that $ G=<B,N>$.
  2. $ B\cap N =T \triangleleft N$ and $ N/T=W$ is a group generated by a set $ S$.
  3. $ sBw \subseteq BwB \cup BswB$ for all $ s\in S$ and $ w\in W$.
  4. $ sBs^{-1} \not\subseteq B$ for all $ s\in S$.

Where $ BwB$ is a double coset with respect to $ B$. It can be proven that $ S$ is in fact made up of elements of order 2, and that $ W$ is a Coxeter group.

Example: Let $ G=GL_n(\mathbb{K})$ where $ \mathbb{K}$ is some field. Then, if we let $ B$ be the subgroup of upper triangular matrices and $ N$ be the subgroup of monomial matrices (i.e. matrices having one nonzero entry in each row and each column, or more precisely the stabilizer of the lines $ \{[e_1],...,[e_n]\}$). Then, it can be shown that $ B$ and $ N$ generate $ G$ and that $ T$ is the subgroup of diagonal matrices. In turn, it follows that $ W$ in this case is isomorphic to the symmetric group on $ n$ letters, $ S_n$.

For more, consult chapter 5 in the book Buildings, by Kenneth Brown



"BN-pair" is owned by tedgar.
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Other names:  Tits Systems
Keywords:  Bruhat decomposition, double cosets, coxeter groups, Lie groups
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Cross-references: symmetric group, isomorphic, diagonal matrices, generate, lines, stabilizer, column, row, matrices, monomial matrices, upper triangular matrices, field, Coxeter group, order, double coset, group generated by, subgroups, group
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This is version 8 of BN-pair, born on 2005-09-07, modified 2005-09-08.
Object id is 7362, canonical name is BNPair.
Accessed 2135 times total.

Classification:
AMS MSC: 20F55 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Reflection and Coxeter groups)
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