Topic BN-Pairs and Tits Systems
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BN-Pairs and Tits Systems

A BN-pair (or Tits system) is an abstraction of the Bruhat decomposition of a reductive group. It is the axiomatic input from which buildings are constructed.

Definition

Let \(G\) be a group. A BN-pair for \(G\) consists of:

  • subgroups \(B, N \subseteq G\);
  • a set \(S \subseteq W := N / (B \cap N)\) of involutions ("simple reflections")

satisfying:

(BN1) \(G = \langle B, N \rangle\);

(BN2) \(B \cap N\) is normal in \(N\), and \(W = N/(B \cap N)\) is generated by \(S\);

(BN3) For all \(s \in S, w \in W\): \(s B w \subseteq B w B \cup B s w B\);

(BN4) For all \(s \in S\): \(s B s^{-1} \not\subseteq B\).

The data \((G, B, N, S)\) is called a Tits system. The group \(W\) is the Weyl group, and \((W, S)\) is automatically a Coxeter system.

The Bruhat decomposition

From the BN-pair axioms one derives:

\[ G \;=\; \bigsqcup_{w \in W} B w B \]

(a disjoint union of double cosets). Each double coset is isomorphic (as a \(B \times B\)-set) to \(B \times U_w\) where \(U_w\) is a specific subgroup. The decomposition is the Bruhat decomposition in its abstract form.

Examples

Reductive groups

For a split connected reductive group \(G\) over a field \(k\) with Borel \(B\), maximal torus \(T \subseteq B\), normaliser \(N = N_G(T)\):

  • \(B \cap N = T\);
  • \(W = N_G(T)/T\) is the Weyl group;
  • \(S\) = simple reflections (reflections through the walls of the fundamental Weyl chamber).

This is the spherical BN-pair, and the resulting building is the spherical building of \(G\).

\(\operatorname{GL}_n\)

For \(G = \operatorname{GL}_n(k)\), \(B\) the upper-triangular Borel, \(N\) the monomial matrices, the Weyl group is \(W = S_n\), the simple reflections are the adjacent transpositions \(s_i = (i, i+1)\). The Bruhat decomposition becomes the LU decomposition of matrices modulo permutation.

\(p\)-adic reductive groups

For \(G\) a connected reductive group over a non-archimedean local field \(F\), there is an affine BN-pair:

  • \(B\) = Iwahori subgroup,
  • \(N\) = normaliser of the maximal split torus,
  • the resulting Weyl group is an affine Weyl group.

The resulting building is the affine building (= Bruhat-Tits building) of \(G\). See affine Weyl groups and [[node:buildings.bruhat_tits_buildings|Bruhat-Tits buildings]].

Parabolic subgroups via BN-pairs

For each subset \(I \subseteq S\), the subgroup \(P_I := B W_I B\) (where \(W_I = \langle I \rangle \le W\)) is a standard parabolic subgroup. All subgroups containing \(B\) are of this form (parabolic-Bruhat correspondence).

Building construction

Tits's general theorem: every BN-pair canonically determines a simplicial complex (the building) on which \(G\) acts, with combinatorial properties controlled by the Coxeter system \((W, S)\).

  • Spherical building: from a spherical BN-pair; the resulting simplicial complex is finite-dimensional and "spherical" in the topological sense (each apartment is a sphere triangulation).
  • Euclidean (affine) building: from an affine BN-pair; apartments are Euclidean spaces tiled by the affine Weyl group.

References

  • Bourbaki, Groupes et Algèbres de Lie IV–VI (1968).
  • Tits, Buildings of Spherical Type and Finite BN-Pairs (LNM 386, 1974).
  • Bruhat–Tits, Groupes réductifs sur un corps local I/II (1972, 1984).

Prerequisite Topics

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