Diagram algebras turn combinatorial pictures into operators. Lines connect tensor positions, diagrams multiply by stacking, closed loops contribute a scalar parameter, and the resulting algebras become centralizers for classical and quantum symmetry groups.
Schur–Weyl duality used permutations to centralize GL(V). If V carries a nondegenerate bilinear form, contractions become available in addition to permutations. The Brauer algebra adds exactly those pairings and centralizes orthogonal or symplectic group actions in tensor space. The Temperley–Lieb algebra is a planar diagram algebra that appears as a quotient or centralizer in rank-one quantum settings.
This guide continues from Schur–Weyl Duality, Quantum Groups and Fusion Categories.
Brauer algebra · Diagram multiplication · Orthogonal/symplectic centralizers · Temperley–Lieb · Quantum sl2 · Practice
Why pairings appear when a form is present
Let V carry a nondegenerate symmetric or alternating bilinear form ⟨·,·⟩ preserved by a group G. Then one can contract two tensor factors using the form.
This operation commutes with the G-action because
⟨gv,gw⟩=⟨v,w⟩.
Permutations are therefore no longer the only natural G-equivariant operators on V^{⊗r}. Pairings and coevaluation-type insertions enlarge the commutant.
Brauer diagrams
A Brauer r-diagram has r vertices on a top row and r on a bottom row, paired into r edges. Edges may connect top to top, bottom to bottom, or top to bottom.
The vector space with these diagrams as a basis becomes the Brauer algebra B_r(δ), where δ is a scalar parameter.
The number of pairings of 2r labelled vertices is
(2r−1)!!=(2r−1)(2r−3)···3·1.
Therefore dim B_r(δ)=(2r−1)!! independent of δ as a vector-space dimension.
Worked dimension counts
For r=1 there is one pairing. For r=2 there are 3 pairings. For r=3 there are 15. For r=4 there are 105.
The rapid growth comes from allowing arbitrary pairings, not only permutations. The symmetric-group algebra C[S_r] has dimension r!, which is smaller than (2r−1)!! once contractions become significant.
Multiplication by stacking
To multiply two Brauer diagrams, place the first above the second and identify the bottom vertices of the first with the top vertices of the second. Trace the strands from the outer top row to the outer bottom row.
Closed loops can appear entirely inside the middle layer. Remove each such loop and multiply the remaining diagram by δ for every removed loop.
Thus if stacking produces k closed loops and outer diagram D, then
D₁D₂=δ^k D.
Associativity follows from the fact that stacking three diagrams gives the same connectivity and total number of internal loops regardless of which pair is multiplied first.
Generators
The Brauer algebra can be generated by permutation-like elements s_i and contraction elements e_i.
- s_i swaps adjacent strands i and i+1;
- e_i pairs the adjacent top vertices and adjacent bottom vertices.
Among the defining relations are
- s_i²=1;
- e_i²=δe_i;
- s_ie_i=e_i=e_is_i;
- the symmetric-group braid relations for s_i;
- additional mixed relations expressing local diagram isotopy.
The equation e_i²=δe_i is the algebraic version of stacking the contraction diagram with itself and producing one closed loop.
From diagrams to tensor operators
Suppose V has a nondegenerate symmetric form and choose dual bases {v_a},{v^a}. A top-to-top pairing contracts two tensor inputs using ⟨·,·⟩. A bottom-to-bottom pairing inserts the invariant tensor
Ω=Σ_a v_a⊗v^a.
Vertical strands transport tensor factors. Combining these local rules assigns a linear endomorphism of V^{⊗r} to every Brauer diagram.
The value of a closed loop becomes dim V in the orthogonal case with standard normalization, because contraction of Ω gives the trace of the identity.
Orthogonal centralizer duality
Let dim V=n and let O(V) preserve a nondegenerate symmetric bilinear form. Then the Brauer algebra B_r(n) acts on V^{⊗r} and commutes with O(V).
In the appropriate stable range, its image is the full commutant:
End_{O(V)}(V^{⊗r})=image B_r(n).
For small n relative to r, the Brauer action can have a kernel, just as the symmetric-group action can fail to be faithful in Schur–Weyl duality.
Thus “Brauer algebra is the centralizer” should usually be read as a statement about its image unless faithfulness has been separately verified.
Symplectic centralizer duality
For a symplectic vector space V of even dimension 2m, the same diagram algebra acts after sign conventions are adjusted to the alternating form.
One common convention uses the parameter δ=−2m for the symplectic action, while alternative conventions absorb signs into the diagram generators and report +2m. The actual operator realization must be compared before identifying parameter values.
The centralizer is the image of a Brauer algebra in End(V^{⊗r}) and pairs the symplectic group with a diagrammatic commuting symmetry.
Why classical invariant theory appears
The first fundamental theorem for orthogonal and symplectic groups says invariant tensors are generated by pairwise contractions. Brauer diagrams encode exactly those pairings.
Centralizer algebras and invariant theory are therefore two views of the same mechanism: contractions generate invariant tensors, and diagrams generated by those contractions generate commuting endomorphisms.
Planarity changes the algebra
Brauer diagrams allow crossings because permutation symmetry is part of the algebra. Temperley–Lieb diagrams forbid crossings.
This restriction dramatically reduces the number of basis diagrams and changes the algebra from general pairings to planar pairings.
The resulting dimensions are Catalan numbers rather than double factorials.
Temperley–Lieb algebra
The Temperley–Lieb algebra TL_r(δ) has a basis of planar noncrossing pairings between r top and r bottom boundary points.
Its standard generators U_i satisfy
- U_i²=δU_i;
- U_iU_{i±1}U_i=U_i;
- U_iU_j=U_jU_i for |i−j|≥2.
The relation U_i²=δU_i again comes from one closed loop. The triple-product relation is a planar isotopy relation.
Catalan dimension
The number of planar noncrossing pairings of 2r boundary points is the Catalan number
C_r=(1/(r+1)) binom(2r,r).
Therefore dim TL_r(δ)=C_r.
The first values are 1,2,5,14,42 for r=1,2,3,4,5.
Compare r=4: Brauer has dimension 105 while Temperley–Lieb has dimension 14. Planarity removes most pairings.
Temperley–Lieb as a Hecke quotient
For type A, TL_r can be obtained as a quotient of the Hecke algebra H_r(q) by the ideal killing the three-strand antisymmetrizer in the rank-one quantum representation setting.
This is the quantum analogue of the fact that Λ³(C²)=0 in ordinary Schur–Weyl theory. Tensor space of a two-dimensional representation cannot support partitions with more than two rows.
The quotient therefore remembers only the portion of the Hecke algebra visible to quantum sl₂ tensor powers.
Quantum sl2 centralizer
Let V be the two-dimensional defining representation of U_q(sl₂) at generic q. On V^{⊗r}, the quantum-group action commutes with a Temperley–Lieb action.
With the usual normalization, the loop parameter is
δ=q+q^{-1}
up to possible sign conventions.
The Temperley–Lieb algebra becomes the centralizer algebra of the generic quantum sl₂ action, again in the appropriate image/stable interpretation.
Classical limit
Set q=1. Then δ=2, matching the ordinary two-dimensional sl₂ representation and the familiar decomposition of tensor powers into spin sectors.
The diagram algebra survives the classical limit, while its parameter records the categorical dimension of the defining object.
At roots of unity, TL_r(δ) can fail to be semisimple. Jones–Wenzl projectors may cease to exist at certain ranks, and quotient categories produce the finite fusion rules seen in SU(2)_k-type theories.
Jones–Wenzl projectors
At generic parameter, the Jones–Wenzl projector f_r is the distinguished idempotent in TL_r that kills every U_i:
U_if_r=0=f_rU_i.
It projects onto the highest-spin summand of the r-fold tensor power in the quantum sl₂ picture.
Recursive formulas involve quantum integers in denominators. At roots of unity, those denominators can vanish, explaining why the semisimple projector calculus breaks at specific levels.
Diagram categories
Instead of fixing r, one can organize all diagram spaces simultaneously into a monoidal category. Objects are nonnegative integers or tensor powers, and morphisms are diagrams.
Horizontal juxtaposition becomes tensor product of morphisms; vertical stacking becomes composition.
This categorical viewpoint makes diagram algebras endomorphism algebras of objects inside a larger tensor category, connecting directly to the Fusion Categories and Categorification routes.
Beyond Brauer and Temperley–Lieb
Centralizer theory produces many related algebras:
- partition algebras for symmetric-group actions on permutation modules;
- walled Brauer algebras for mixed tensor powers V^{⊗r}⊗(V*)^{⊗s};
- BMW algebras for quantum orthogonal/symplectic settings;
- Hecke algebras for quantum GL_n;
- Temperley–Lieb algebras for rank-one quotients.
The recurring design is the same: identify a tensor-space symmetry, then compute the algebra of operators commuting with it.
A verification workflow
- 1. Identify the preserved form or tensor.
- 2. Determine which contractions are equivariant.
- 3. Translate local operations into diagrams.
- 4. State the loop parameter convention.
- 5. Verify e_i²=δe_i or U_i²=δU_i diagrammatically.
- 6. Distinguish crossings allowed from forbidden.
- 7. Check the dimension: double factorial for Brauer, Catalan for TL.
- 8. Treat centralizers as images unless faithfulness is known.
- 9. At quantum parameters, state whether q is generic or a root of unity.
- 10. Do not import semisimple projector formulas across singular parameters.
Common mistakes
- Confusing Brauer and braid: Brauer diagrams pair endpoints; braid diagrams track crossings without contractions.
- Forgetting closed-loop factors.
- Calling every pairing planar: Brauer allows crossings; TL does not.
- Using δ=n in every symplectic convention.
- Assuming diagram actions are faithful in all dimensions.
- Equating Catalan and double-factorial basis counts.
- Applying generic Jones–Wenzl recursion at roots where denominators vanish.
- Confusing the diagram parameter δ with vector-space dimension when the category is deformed.
Practice questions
1. Find dim B_3(δ). 2. Find dim TL_4(δ). 3. What does a closed loop contribute? 4. Why do contractions commute with O(V)?
5. What is the local meaning of e_i²=δe_i? 6. What group is paired with the Brauer algebra in the symmetric-form case? 7. What is the generic quantum sl₂ loop parameter? 8. Why is TL a quotient of a Hecke algebra in rank two?
9. What happens to Jones–Wenzl projectors at certain roots of unity? 10. Distinguish Brauer from Temperley–Lieb in one sentence. 11. What algebra handles mixed V and V* tensor powers? 12. Why should centralizer statements often say “image”?
Worked answers
1. (2·3−1)!!=5!!=15.
2. C_4=14.
3. A factor δ.
4. The bilinear form is invariant, so contracting gv and gw gives the same scalar as contracting v and w.
5. Stacking e_i with itself creates one removable closed loop and leaves e_i.
6. The orthogonal group O(V).
7. δ=q+q^{-1}, up to sign convention.
8. The rank-two tensor representation kills the three-strand antisymmetrizer, producing the TL quotient.
9. Recursion denominators can vanish and the generic idempotents can cease to exist.
10. Brauer allows arbitrary pairings/crossings; Temperley–Lieb retains only planar noncrossing pairings.
11. The walled Brauer algebra.
12. Small tensor-space dimension can create a kernel in the abstract diagram algebra action.
Sources and further study
Richard Brauer’s original centralizer work introduced the Brauer algebra for orthogonal and symplectic groups. Standard modern references include Goodman and Wallach, Symmetry, Representations, and Invariants; Paul Martin, Potts Models and Related Problems in Statistical Mechanics; and Frederick Goodman, Pierre de la Harpe and Vaughan Jones, Coxeter Graphs and Towers of Algebras. For the quantum route, continue through the BTT Quantum Groups and Fusion Categories guides.
Representation Mathematics — Batch 09
Begin with permutation centralizers in Schur–Weyl Duality. Build quantum groups by counting extensions in Hall Algebras. Follow symmetric-group sequences through Representation Stability and FI-Modules. Return to the BTT Mathematics Learning Hub.