07.04.10 · representation-theory / symmetric

Bruhat decomposition

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Anchor (Master): Bruhat 1954 Bull. Soc. Math. France; Tits 1974 Buildings of Spherical Type; Humphreys 1990 Reflection Groups and Coxeter Groups

Intuition [Beginner]

The Bruhat decomposition partitions a Lie group $G$ (like $S L (n, R)$ ) into pieces labelled by the Weyl group $W$ . Each piece is a "double coset" $B w B$ : you multiply a Borel subgroup $B$ (the upper-triangular matrices, roughly) on the left by $B$ , then insert a Weyl group element $w$ (a permutation matrix), then multiply by $B$ again on the right.

For $G = S L (n, R)$ , the Borel subgroup $B$ is the group of upper-triangular matrices with determinant 1, and the Weyl group $W$ is the symmetric group $S_{n}$ (represented by permutation matrices). The decomposition says that every matrix in $S L (n, R)$ can be written as $b_{1} w b_{2}$ where $b_{1}, b_{2}$ are upper-triangular and $w$ is a permutation matrix. The permutation $w$ is uniquely determined by the matrix.

The Weyl group element $w$ measures how "mixed up" the columns of the matrix are relative to the flag (nested subspaces) fixed by $B$ . The identity element corresponds to matrices already in $B$ (upper-triangular). The longest element $w_{0}$ (the full reversal permutation) corresponds to the "most mixed" piece.

Why does this concept exist? The Bruhat decomposition is the combinatorial skeleton of the group. It governs the geometry of flag varieties (via Schubert cells), the representation theory (via the Borel-Weil theorem), and the structure of the spherical Hecke algebra. It is to Lie groups what the Jordan normal form is to linear algebra: a canonical decomposition into elementary pieces.

Visual [Beginner]

A diagram showing the Bruhat decomposition of $S L (2, R)$ into two cells: the "big cell" $B \cdot e \cdot B = B$ (upper-triangular matrices, dimension 2) and the "small cell" $B \cdot s \cdot B$ (where $s = (01 - 1 0)$ is the Weyl reflection, dimension 2). The group $S L (2, R)$ (dimension 3) is shown as a union of these two pieces.

The two cells fit together like an open dense set and its complement, giving a stratification of the group by Weyl group elements.

Worked example [Beginner]

Determine the Bruhat cell of $g = (0110)$ in $S L (2, R)$ .

Step 1. The Borel subgroup is $B$ = upper-triangular matrices with determinant 1. The Weyl group has two elements: $e$ and $s$ (the swap/reflection).

Step 2. Check if $g \in B$ : the $(2, 1)$ entry is 1, but $B$ requires the $(2, 1)$ entry to be 0. So $g \in / B = B e B$ .

Step 3. Verify $g \in B s B$ . Set $b_{1} = (1001)$ and $b_{2} = (1001)$ . Use the Weyl group representative $s = (0110)$ (the permutation matrix that swaps rows). Then $b_{1} s b_{2} = s = g$ .

What this tells us: $g$ is itself a Weyl group element (the non-identity one), so $g \in B s B$ . The cell $B s B$ consists of all matrices in $S L (2, R)$ with nonzero $(2, 1)$ entry. The two Bruhat cells are $B$ (upper-triangular, $g_{21} = 0$ ) and $B s B$ ( $g_{21} \neq = 0$ ).

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $G$ be a reductive algebraic group over a field $k$ (or a real semisimple Lie group). Fix a Borel subgroup $B$ and a maximal torus $T \subset B$ .

Definition (Weyl group). The Weyl group $W = N_{G} (T) / T$ is the quotient of the normaliser of $T$ by $T$ itself. It is a finite Coxeter group generated by simple reflections $s_{1}, \dots, s_{r}$ corresponding to the simple roots.

Definition (Bruhat decomposition). The group $G$ decomposes as a disjoint union:

$G = w \in W ⨆ B w B$

where each Bruhat cell $B w B = {b_{1} w b_{2} : b_{1}, b_{2} \in B}$ is a locally closed subvariety (or submanifold) of $G$ .

Definition (Bruhat order). The Bruhat order on $W$ is defined by: $v \leq w$ if and only if $B v B \subset \overline{B w B}$ (the closure of the Bruhat cell). Equivalently, $v \leq w$ if for every reduced expression $w = s_{i_{1}} \dots s_{i_{k}}$ , there exists a subexpression $v = s_{i_{j_{1}}} \dots s_{i_{j_{m}}}$ obtained by deleting some factors.

Definition (Length function). The length $ℓ (w)$ is the minimum number of simple reflections in an expression for $w$ . Equivalently, $ℓ (w) = dim (B w B) - dim (B)$ .

Counterexamples to common slips

The Bruhat decomposition depends on the choice of Borel subgroup, but the decomposition as a partition is canonical up to conjugacy. Any two Borel subgroups are conjugate, so different choices of $B$ give the same partition up to relabelling by an inner automorphism.
The closure $\overline{B w B}$ is not a Bruhat cell in general. It is a union of Bruhat cells: $\overline{B w B} = ⨆_{v \leq w} B v B$ . The closure relation is governed by the Bruhat order.
Not every $g \in G$ can be written as $g = b_{1} w b_{2}$ with a unique $b_{1}, b_{2} \in B$ . The Weyl group element $w$ is unique, but $b_{1}$ and $b_{2}$ are not (they are determined up to the action of $T = B \cap w B w^{- 1}$ ).

Key theorem with proof [Intermediate+]

Theorem (Bruhat decomposition). Let $G$ be a connected reductive group with Borel subgroup $B$ , maximal torus $T \subset B$ , and Weyl group $W = N_{G} (T) / T$ . Then $G = ⨆_{w \in W} B w B$ (disjoint union).

Proof.

Step 1 (Setup for $G L (n)$ ). We first prove the result for $G = G L (n, k)$ with $B$ the upper-triangular matrices and $T$ the diagonal matrices. The Weyl group $W = S_{n}$ acts by permutation matrices.

Step 2 (Column echelon form). For any $g \in G L (n, k)$ , consider the columns $v_{1}, \dots, v_{n}$ of $g$ . Since $g$ is invertible, these are linearly independent. Right multiplication by $B$ (upper-triangular) replaces $v_{j}$ by a linear combination $c_{1 j} v_{1} + \dots + c_{j j} v_{j}$ with $c_{j j} \neq = 0$ (since $B$ is upper-triangular). So right multiplication by $B$ performs column operations that add multiples of earlier columns to later columns and rescale.

Left multiplication by $B$ performs analogous row operations. Together, $b_{1} g b_{2}$ performs: row operations that add multiples of earlier rows to later rows (left multiplication by $B$ ), and column operations that add multiples of earlier columns to later columns (right multiplication by $B$ ).

Step 3 (Permutation matrix reduction). For each $g$ , perform the following: scan the columns left to right. For column $j$ , find the first row $i$ such that the $(i, j)$ entry can be made nonzero by row operations (adding multiples of earlier rows). By suitable row operations (left multiplication by $B$ ), make all entries above position $(i, j)$ in column $j$ equal to zero, and make the $(i, j)$ entry equal to 1.

The permutation $w (j) = i$ records the pivot positions. By construction, $g = b_{1} w b_{2}$ where $w$ is the permutation matrix with 1s in positions $(w (j), j)$ and $b_{1}, b_{2} \in B$ . This shows $G = B W B$ (as a set union).

Step 4 (Disjointness). Suppose $B w B = B w^{'} B$ for $w \neq = w^{'}$ . Then $w^{'} = b_{1} w b_{2}$ for some $b_{1}, b_{2} \in B$ . The permutation matrix $w^{'}$ has a single 1 in each row and column. The product $b_{1} w b_{2}$ has the property that its "pivots" (first nonzero entries in each column, reading from top to bottom) are in the same positions as those of $w$ (since left multiplication by $B$ only adds multiples of earlier rows, preserving the pivot structure of $w$ , and right multiplication by $B$ adds multiples of earlier columns). But the pivots of $w^{'}$ are determined by the permutation $w^{'}$ , and the pivots of $b_{1} w b_{2}$ are determined by $w$ . Since different permutations have different pivot structures, $w = w^{'}$ .

Step 5 (General reductive groups). For a general reductive group $G$ , choose a split maximal torus $T \subset B$ . The Bruhat decomposition is proved by reduction to rank one: for each simple reflection $s_{α}$ , the group $G_{α} = ⟨ B, s_{α} B s_{α} ⟩$ is isomorphic to $S L (2, k)$ or $P G L (2, k)$ (modulo the center), and the Bruhat decomposition for $S L (2)$ gives $G_{α} = B_{α} ⊔ B_{α} s_{α} B_{α}$ .

For the general case, any $g \in G$ can be reduced by induction on the length of the Weyl group element. Starting from $g = b_{1} w b_{2}$ (existence follows from the BN-pair axioms), the Weyl group element is determined by the coset $B g B \in B \ G / B$ . The BN-pair axioms guarantee that $B \ G / B ≅ W$ , giving the decomposition $G = ⨆_{w \in W} B w B$ . $□$

Bridge. The Bruhat decomposition builds toward the representation theory of reductive groups via the Borel-Weil theorem, where line bundles on the flag variety $G / B$ realise the irreducible representations. The foundational reason is that the Bruhat cells $B w B$ give a cell decomposition of $G / B$ into Schubert cells $X_{w} = B w B / B$ , and this is exactly the structure that determines the cohomology of the flag variety. The bridge is the identification of the Bruhat order with the closure relations among Schubert cells, and putting these together with the Iwasawa decomposition of 07.04.09, the Borel subgroup $B = A N$ indexes the same subgroup in both decompositions. The Bruhat decomposition generalises the row reduction algorithm from linear algebra to arbitrary reductive groups.

Exercises [Intermediate+]

Exercise 3 (medium, proof).

For $G = S L (2, R)$ , prove that the Bruhat decomposition has exactly two cells: $B$ and $B s B$ where $s = (0 - 1 10)$ .

Hint

For any $g \in S L (2, R)$ with $g_{21} = 0$ , show $g \in B$ . For $g_{21} \neq = 0$ , show $g \in B s B$ .

Answer

The Borel subgroup is $B = {(a 0 b a^{- 1}) : a \neq = 0, b \in R}$ (upper-triangular).

Case 1: $g_{21} = 0$ . Then $g = (g_{11} 0 g_{12} g_{22})$ with $g_{11} g_{22} = 1$ (determinant 1). So $g \in B$ .

Case 2: $g_{21} \neq = 0$ . Compute $g \cdot s^{- 1} = g (01 - 1 0) = (g_{12} g_{22} - g_{11} - g_{21})$ . The $(2, 1)$ entry is $g_{22}$ . If $g_{22} \neq = 0$ , then by suitable left multiplication by $B$ (adding $- g_{22} / g_{21}$ times row 2 to row 1, then rescaling), we can make $g s^{- 1}$ upper-triangular: $g s^{- 1} \in B$ , so $g \in B s B$ . If $g_{22} = 0$ , then $g_{11} = - g_{21}^{- 1}$ (from $det g = 1$ and $g_{22} = 0$ ), and a similar row reduction works.

In both cases, $g \in B$ or $g \in B s B$ , and $B \cap B s B = \emptyset$ (since $B$ has $g_{21} = 0$ and $B s B$ has $g_{21} \neq = 0$ ).

Exercise 4 (medium, proof).

For $G = S L (n, R)$ , show that the Bruhat cell $B w B$ consists of all matrices $g$ whose "pivots" (first nonzero entry in each column, reading top to bottom) are in the positions $(w (1), 1), (w (2), 2), \dots, (w (n), n)$ .

Hint

Right multiplication by $B$ adds multiples of earlier columns to later columns. Left multiplication by $B$ adds multiples of earlier rows to later rows. Track how these operations change the pivot positions.

Answer

Let $g \in B w B$ , so $g = b_{1} w b_{2}$ . The permutation matrix $w$ has 1s in positions $(w (j), j)$ and 0s elsewhere. Its pivots are at positions $(w (j), j)$ by construction.

Right multiplication by $b_{2} \in B$ (upper-triangular): this replaces column $j$ by $c_{1 j} v_{1} + \dots + c_{j j} v_{j}$ where $c_{j j} \neq = 0$ . The pivot in column $j$ (first nonzero entry from top) remains at row $w (j)$ because $c_{j j} \neq = 0$ ensures the entry at $(w (j), j)$ remains nonzero, and adding multiples of earlier columns does not introduce nonzero entries above row $w (j)$ in column $j$ (since earlier columns have pivots at rows $w (1), \dots, w (j - 1)$ which may be above or below $w (j)$ ).

Actually, this is not quite right for general $w$ . The correct statement uses the "rank profile": for each $j$ , the rank of the first $j$ columns of $g$ equals the rank of the first $j$ columns of $w$ , and the pivot in column $j$ is at row $w (j)$ . This characterises the Bruhat cell.

Exercise 5 (medium, proof).

Show that $dim (B w B) = dim (B) + ℓ (w)$ where $ℓ (w)$ is the length of $w$ in the Weyl group (number of inversions for $S_{n}$ ).

Hint

The map $B \times B \to B w B$ given by $(b_{1}, b_{2}) \mapsto b_{1} w b_{2}$ is surjective but not injective. Compute the dimension by counting the fibre.

Answer

The map $φ_{w} : B \times B \to G$ given by $(b_{1}, b_{2}) \mapsto b_{1} w b_{2}$ has image $B w B$ . Its fibre over $w$ is ${(b_{1}, b_{2}) : b_{1} w b_{2} = w} = {(b, w^{- 1} b^{- 1} w) : b \in B \cap w B w^{- 1}}$ . So $dim (B w B) = 2 dim (B) - dim (B \cap w B w^{- 1})$ .

The group $B \cap w B w^{- 1}$ contains $T$ (since $T \subset B$ and $w T w^{- 1} = T$ ). In fact, $B \cap w B w^{- 1}$ is generated by $T$ and the root subgroups $U_{α}$ for $α > 0$ such that $w α > 0$ . The number of positive roots sent to negative roots by $w$ is $ℓ (w)$ . So $dim (B \cap w B w^{- 1}) = dim (T) + (∣ Φ^{+} ∣ - ℓ (w))$ , and $dim (B) = dim (T) + ∣ Φ^{+} ∣$ .

Therefore $dim (B w B) = 2 (dim (T) + ∣ Φ^{+} ∣) - (dim (T) + ∣ Φ^{+} ∣ - ℓ (w)) = dim (T) + ∣ Φ^{+} ∣ + ℓ (w) = dim (B) + ℓ (w)$ .

Exercise 6 (hard, proof).

Prove that the big cell $B w_{0} B$ (where $w_{0}$ is the longest element of $W$ ) is open and dense in $G$ .

Hint

$ℓ (w_{0}) = ∣ Φ^{+} ∣$ (all positive roots are sent to negative roots). So $dim (B w_{0} B) = dim (B) + ∣ Φ^{+} ∣ = dim (G)$ .

Answer

The dimension of $B w_{0} B$ is $dim (B) + ℓ (w_{0}) = dim (B) + ∣ Φ^{+} ∣$ (since $w_{0}$ sends all positive roots to negative roots, giving $ℓ (w_{0}) = ∣ Φ^{+} ∣$ ). Now $dim (B) = dim (T) + ∣ Φ^{+} ∣$ (the torus plus the positive root spaces). So $dim (B w_{0} B) = dim (T) + 2∣ Φ^{+} ∣$ .

For a semisimple group $G$ , $dim (G) = dim (T) + ∣ Φ^{+} ∣ + ∣ Φ^{-} ∣ = dim (T) + 2∣ Φ^{+} ∣$ . So $dim (B w_{0} B) = dim (G)$ .

Since $B w_{0} B$ is a constructible set of full dimension in the irreducible variety $G$ , it contains a Zariski-open subset of $G$ . Being a single $B \times B$ -orbit, it is smooth and locally closed. By dimension counting, $B w_{0} B$ is open and dense.

Exercise 7 (hard, proof).

For $G = S L (n, R)$ , prove that the Bruhat order on $S_{n}$ is characterised by: $σ \leq τ$ if and only if for all $1 \leq i \leq j \leq n$ , the number of values $σ (k) \leq i$ with $k \leq j$ is at least the number of values $τ (k) \leq i$ with $k \leq j$ .

Hint

This is the "rank condition" characterisation. The condition says that the submatrix formed by the first $j$ columns and first $i$ rows of the permutation matrix has rank at least as large for $σ$ as for $τ$ .

Answer

The rank condition: for any $i, j$ , let $r_{σ} (i, j) = ∣ {k \leq j : σ (k) \leq i} ∣$ be the number of 1s in the submatrix of the permutation matrix of $σ$ formed by rows $1, \dots, i$ and columns $1, \dots, j$ . This equals the rank of that submatrix.

The Bruhat order is $σ \leq τ$ iff $r_{σ} (i, j) \geq r_{τ} (i, j)$ for all $i, j$ (equivalently: the rank of each upper-left submatrix for $σ$ is at most that for $τ$ ).

Forward direction: if $σ \leq τ$ (i.e., $\overline{B σ B} \supset B τ B$ ), then for any representative matrix $g_{τ}$ in $B τ B$ , there is a sequence of matrices in $B σ B$ converging to $g_{τ}$ . The rank function is lower semicontinuous: $r_{g} (i, j) \geq r_{g_{τ}} (i, j)$ for nearby $g$ . Since matrices in $B σ B$ have $r_{g} (i, j) = r_{σ} (i, j)$ , we get $r_{σ} (i, j) \geq r_{τ} (i, j)$ .

Reverse direction: if $r_{σ} (i, j) \geq r_{τ} (i, j)$ for all $i, j$ , then by Gaussian elimination, any matrix with the pivot structure of $τ$ can be deformed to have the pivot structure of $σ$ , showing $B τ B \subset \overline{B σ B}$ .

Advanced results [Master]

Theorem 1 (Schubert cells and flag varieties). The flag variety $G / B$ decomposes into Schubert cells $X_{w} = B w B / B$ , each isomorphic to affine space $A^{ℓ (w)}$ . The closure $\overline{X_{w}} = ⨆_{v \leq w} X_{v}$ is the Schubert variety, whose cohomology class $[X_{w}]$ is the Schubert class in $H^{2 (d i m G / B - ℓ (w))} (G / B)$ .

Theorem 2 (Bruhat order and Coxeter groups). The Bruhat order on a Coxeter group $(W, S)$ is determined by the reduced expressions: $v \leq w$ iff there exists a reduced expression $w = s_{i_{1}} \dots s_{i_{k}}$ and a subsequence $v = s_{i_{j_{1}}} \dots s_{i_{j_{m}}}$ obtained by deleting factors. The Bruhat order is graded by the length function.

Theorem 3 (BN-pair axioms). The pair $(B, N)$ forms a Tits system (BN-pair) satisfying: (BN1) $G$ is generated by $B$ and $N$ ; (BN2) $H = B \cap N$ is normal in $N$ ; (BN3) $W = N / H$ is generated by a set $S$ of involutions; (BN4) $s B w \subset B w B \cup B s w B$ for all $s \in S$ , $w \in W$ ; (BN5) $s B s \neq = B$ for all $s \in S$ . The Bruhat decomposition follows from these axioms.

Theorem 4 (Chevalley formula). The product of a Schubert class $[X_{w}]$ with a divisor class $[X_{s}]$ (for a simple reflection $s$ ) in the cohomology ring $H^{*} (G / B)$ is given by the Chevalley formula: $[X_{s}] \cdot [X_{w}] = \sum [X_{v}]$ where the sum is over $v = w s^{'}$ with $ℓ (v) = ℓ (w) + 1$ and $s^{'} \leq w^{- 1} s w$ in the Bruhat order, with coefficients determined by the root data.

Theorem 5 (Kazhdan-Lusztig theory). The Kazhdan-Lusztig polynomials $P_{v, w} (q)$ for $v \leq w$ in the Bruhat order encode the intersection cohomology of Schubert varieties. The coefficients of $P_{v, w}$ are non-negative integers, and $P_{v, w} (1) = 1$ for all $v \leq w$ .

Synthesis. The Bruhat decomposition is the foundational reason that the geometry of flag varieties decomposes into elementary affine pieces (Schubert cells), each indexed by the Weyl group. The central insight is that the double coset decomposition $G = ⨆ B w B$ identifies the group structure with the combinatorial structure of the Coxeter group, and this is exactly the bridge between algebraic geometry and combinatorics. Putting these together with the BN-pair axioms, the Bruhat decomposition axiomatises the structure of reductive groups, and the bridge is the Tits building whose apartments are indexed by the Weyl group.

The pattern generalises: the Schubert calculus on $G / B$ computes intersection numbers via the Bruhat order, and appears again in the Kazhdan-Lusztig conjecture where the polynomials $P_{v, w}$ determine the characters of irreducible representations. The Iwasawa decomposition of 07.04.09 provides the analytic realisation of the Borel subgroup $B = A N$ , and the restricted root system of 07.04.08 determines the Weyl group that indexes the Bruhat cells. The Bruhat decomposition identifies the group-theoretic structure of $G$ with the combinatorial geometry of $W$ , and this is exactly the structure that underlies the Langlands programme.

Full proof set [Master]

Proposition 1 (BN-pair implies Bruhat decomposition). Let $(B, N)$ be a BN-pair in $G$ with Weyl group $W = N / (B \cap N)$ . Then $G = ⨆_{w \in W} B w B$ .

Proof. By axiom (BN1), $G = ⟨ B, N ⟩ = B W B$ (as a set). For disjointness, we use (BN4): $s B w \subset B w B \cup B s w B$ for all $s \in S$ , $w \in W$ .

Induction on length. For $ℓ (w) = 0$ (i.e., $w = e$ ): $B e B = B$ . For $ℓ (w) = 1$ (simple reflections): by (BN5), $s B \neq \subset B$ (since $s B s \neq = B$ implies $s B \neq = B$ ), so $s B \cap B s w B \neq = \emptyset$ by (BN4), giving $s B \subset B \cup B s B$ and $B s B \neq = \emptyset$ .

For general $w = s_{i_{1}} \dots s_{i_{k}}$ (reduced): by induction, $B w \subset B s_{i_{1}} B s_{i_{2}} \dots s_{i_{k}}$ . By (BN4), $B s_{i_{1}} B \subset B \cup B s_{i_{1}} B$ , so $B w B \subset B s_{i_{2}} \dots s_{i_{k}} B \cup B s_{i_{1}} s_{i_{2}} \dots s_{i_{k}} B$ . Continuing by induction on $k$ gives $B w B \subset ⋃_{v \leq w} B v B$ where $v$ ranges over subexpressions.

For uniqueness: suppose $w = b_{1} w^{'} b_{2}$ for $b_{1}, b_{2} \in B$ . Then $w^{'} \in B w B$ . By the exchange axiom (which follows from BN4), this implies $w^{'} = w$ (since both $w, w^{'}$ are in reduced form and $B w B = B w^{'} B$ forces $w = w^{'}$ by the unique coset property). $□$

Proposition 2 (Dimension of Schubert cells). The Schubert cell $X_{w} = B w B / B$ in $G / B$ has dimension $ℓ (w)$ .

Proof. The map $B \to B w B / B$ given by $b \mapsto b w B / B = b w B$ is surjective with fibre $B \cap w B w^{- 1}$ . As computed in Exercise 5: $dim (B w B) = dim (B) + ℓ (w)$ . Since $dim (B w B / B) = dim (B w B) - dim (B)$ , we get $dim (X_{w}) = ℓ (w)$ .

Equivalently, the tangent space to $X_{w}$ at $w B$ is $⨁_{α > 0, w α < 0} g_{α}$ (the root spaces for positive roots sent to negative by $w$ ). The number of such roots is $ℓ (w)$ . $□$

Connections [Master]

Iwasawa decomposition 07.04.09. The Borel subgroup $B$ in the Bruhat decomposition coincides with $A N$ from the Iwasawa decomposition (for the split real form). The Iwasawa decomposition gives the diffeomorphism $G ≅ K \times A N$ , while the Bruhat decomposition refines the $A N$ part into Weyl-group-indexed cells.
Restricted root system 07.04.08. The Weyl group $W$ that indexes the Bruhat cells is the same as the Weyl group of the restricted root system. The length function $ℓ (w)$ equals the number of positive roots sent to negative by $w$ , which is read off from the restricted root system.
Cartan involution 07.04.03. The Cartan involution $θ$ provides the maximal compact subgroup $K$ that stabilises the symmetric space. The Bruhat decomposition of $G$ restricts to a decomposition of $K$ -orbits on $G / B$ , giving the real analogue of the Schubert cell decomposition. The compact subgroup $K$ acts transitively on the open Bruhat cell $B w_{0} B$ .

Historical & philosophical context [Master]

Francois Bruhat established the double coset decomposition $G = ⨆_{w \in W} B w B$ in his 1954 paper on representations of locally compact groups (Bull. Soc. Math. France 82) ^[Bruhat1954]. Bruhat's original context was the study of induced representations and the distribution characters of $p$ -adic groups, where the double coset decomposition provides the combinatorial framework for the Hecke algebra.

Jacques Tits introduced the BN-pair axioms in the 1960s and developed the theory of buildings (Buildings of Spherical Type, Lecture Notes in Mathematics 386, 1974) ^[Tits1974], showing that the Bruhat decomposition is an instance of a general combinatorial structure (the building) that underlies all reductive groups. Humphreys' Linear Algebraic Groups (1975) and Reflection Groups and Coxeter Groups (1990) ^{[Humphreys1990]} gave the modern algebraic-group treatment.

Bibliography [Master]

@article{Bruhat1954,
  author = {Bruhat, Francois},
  title = {Representations des groupes localement compacts},
  journal = {Bull. Soc. Math. France},
  volume = {82},
  year = {1954},
  pages = {113--136},
}

@book{Tits1974,
  author = {Tits, Jacques},
  title = {Buildings of Spherical Type and Finite {BN}-Pairs},
  publisher = {Springer},
  year = {1974},
  series = {Lecture Notes in Mathematics},
  volume = {386},
}

@book{Humphreys1975,
  author = {Humphreys, James E.},
  title = {Linear Algebraic Groups},
  publisher = {Springer},
  year = {1975},
}

@book{Humphreys1990,
  author = {Humphreys, James E.},
  title = {Reflection Groups and Coxeter Groups},
  publisher = {Cambridge University Press},
  year = {1990},
}

@book{Knapp2002,
  author = {Knapp, Anthony W.},
  title = {Lie Groups Beyond an Introduction},
  publisher = {Birkhauser},
  year = {2002},
  edition = {2nd},
}