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Group Algebras ​

The group algebra has a basis of the group's elements and the group's own multiplication as its product, extended bilinearly. It is the most elementary construction in this section — which is why it comes last rather than first. The interesting part is not the product. It is the centre.

is commutative exactly when is, which is usually false. But its centre is always commutative, and it always has a basis of class sums: one element per conjugacy class, obtained by adding that class up. So a non-commutative algebra carries a canonical commutative subalgebra, whose dimension is the number of conjugacy classes — which is also the number of irreducible characters, and the doorway to representation theory.

Commutative or not

Cyclic groups are abelian; dihedral ones past n = 2 are not.

source
<notatio-cell value="GroupIsAbelian(CyclicGroup(6))" />
<notatio-cell value="GroupIsAbelian(DihedralGroup(4))" />
<notatio-cell value="GroupOrder(DihedralGroup(4))" />

The class sums span the centre ​

A commutative subalgebra of a non-commutative algebra

A single reflection is not central; the class sum containing it is. The first class is always the identity alone.

source
<notatio-cell value="ClassSum(DihedralGroup(3), 2)" />
<notatio-cell value="IsCentral(DihedralGroup(3), ClassSum(DihedralGroup(3), 2))" />
<notatio-cell value='IsCentral(DihedralGroup(3), GroupBasis("s0"))' />

The class sums have disjoint supports, so they are linearly independent, so counting them counts the centre's dimension. The tests check that against the closed forms: classes for , for with odd, for even.

Counting classes

Z₆ is abelian so every element is its own class. D₃ ≅ S₃ has three; D₄ has five.

source
<notatio-cell value="GroupCentreDimension(CyclicGroup(6))" />
<notatio-cell value="GroupCentreDimension(DihedralGroup(3))" />
<notatio-cell value="GroupCentreDimension(DihedralGroup(4))" />
<notatio-cell value="ConjugacyClasses(DihedralGroup(3))" />

k[Zₙ] is k[x]/(xⁿ−1) ​

For a cyclic group the product just adds indices mod — so is a polynomial ring modulo a single relation. When the base field has -th roots of unity it splits into copies of the field, which is the same CRT-flavoured splitting the finite hypercomplex page and the residue numerals keep running into.

Adding indices

2 + 5 = 7 ≡ 1. And (1 + x)(1 + x⁵) = 2 + x + x⁵.

source
<notatio-cell value='GroupProduct(CyclicGroup(6), GroupBasis("2"), GroupBasis("5"))' />
<notatio-cell value="GroupOrder(GroupDirectProduct(CyclicGroup(2), CyclicGroup(3)))" />

The dihedral relations ​

Elements are written k for and s k for , with , and .

s² = 1 and s r s = r⁻¹

s² is the identity, and conjugating r by s inverts it — r³ in D₄.

source
<notatio-cell value='GroupProduct(DihedralGroup(4), GroupBasis("s0"), GroupBasis("s0"))' />
<notatio-cell value='GroupProduct(DihedralGroup(4), GroupProduct(
                                             DihedralGroup(4),
                                             GroupBasis("s0"),
                                             GroupBasis("1"),
                                           ), GroupBasis("s0"))' />

Things worth knowing ​

The group axioms are verified, not assumed. The multiplication tables are checked for associativity, an identity, inverses, and the Latin-square property on every group here — because everything downstream rests on them being groups at all.

Several routes to the same algebra. is also the symmetric-group diagram algebra, and also at . Three libraries, one object.

GroupProduct takes its group, since a basis element carries no reference to the group it came from — the same shape as QuiverCompose.

Not built yet. Character theory: the character table, orthogonality relations, and the decomposition of into matrix blocks. That is what the centre is the doorway to, and it is the obvious next step. Also only cyclic, dihedral and direct products — permutation groups come in through the diagram and Hecke libraries instead.