Modular representation theory

Modular representation theory studies representations of a finite group $G$ over a field whose characteristic divides $|G|$. The group algebra is then generally nonsemisimple, and projective modules, radicals, blocks and vertices replace complete reducibility as central tools.

Let $C_{p^n}=⟨g⟩$ and let $k$ have characteristic $p$. With $u=g-1$,Its finite-dimensional indecomposable modules areEach is a uniserial module, with radical $u{M}_r$ and one-dimensional socle $u^{r-1}{M}_r$.

If $G$ is a finite $p$-group and $k$ has characteristic $p$, then the augmentation ideal is the Jacobson radical of $kG$ and $kG/J(kG)\cong k$. Thus $kG$ is a local ring, has only the trivial simple module, and its regular module is indecomposable.

A p-modular system consists of a characteristic-zero field $K$, a discrete valuation ring $\mathcal{O}\subset K$, and its residue field $k$ of characteristic $p$. It is a splitting system for a finite group when both $K$ and $k$ split the relevant group representations.

A p-regular element of a finite group is an element whose order is coprime to $p$. It is also called a $p^{\prime }$-element. The complementary elements are p-singular.

Let $M$ be a representation over a splitting field $k$ of characteristic $p$. For a p-regular element $g$, the eigenvalues of $g$ on $M$ have order prime to $p$. Lift them to characteristic-zero roots of unity by the Teichmuller lift; their sum is the Brauer character value ${\chi }_M(g)$.

The Brauer character of a finite-dimensional modular representation determines its semisimplification. More strongly, the irreducible Brauer characters are linearly independent as complex-valued functions on the p-regular conjugacy classes.

The Brauer character of a projective module over a group algebra is called a projective character. It is the restriction to p-regular elements of the ordinary character of a lifted projective lattice; that ordinary character vanishes on p-singular elements.

For class functions on the p-regular elements of a finite group,Equivalently, summing over p-regular conjugacy-class representatives $x$ gives weights $1/|C_G(x)|$.

If $S_i$ are the simple modules of a split group algebra and $P_i$ are their projective covers, then

For p-regular elements $g,h$,where $S$ ranges over the simple modules and $P_S$ is the projective cover of $S$.

A finite-dimensional algebra $A$ is symmetric when there is a nondegenerate symmetric bilinear form $B:A\times A\to k$ satisfying $B(ab,c)=B(a,bc)$. Equivalently, $A\cong A^{\ast }$ as $A$-bimodules.

For a finite group $G$, the coefficient of the identity in $xy$ defines a nondegenerate symmetric associative bilinear form on $kG$. Hence $kG\cong (kG{)}^{\ast }$ as bimodules and the group algebra is a symmetric algebra.

For finite-dimensional modules over $kG$, projectivity and injectivity are equivalent. Indeed, $(kG{)}^{\ast }$ is injective because ${\mathrm{Hom}}_{kG}(-,(kG{)}^{\ast })\cong {\mathrm{Hom}}_k(-,k)$ is exact, while symmetry gives $(kG{)}^{\ast }\cong kG$.

If $P$ is an indecomposable projective $kG$-module, then its head and socle are simple and naturally isomorphic:This is the identity Nakayama permutation of the symmetric algebra $kG$.

For every simple finite-dimensional $kG$-module $S$,

Let $M$ be an indecomposable finite-dimensional $kG$-module and let $P_k$ be the projective cover of the trivial module. Then

For a subgroup $H\le G$, an $RG$-module $M$ is relatively H-projective when every $RG$-epimorphism onto $M$ that splits after restriction to $H$ already splits over $G$. Equivalently, $M$ is a direct summand of

For $\alpha \in {\mathrm{End}}_{RH}(M)$, the relative trace is

Let a finite group $G$ act on an algebra $A$ by conjugation and let $H\le G$. The transfer ideal from $H$ isThe notation records both the source subgroup and the ambient fixed-point algebra.

D. Higman's criterion says that an $RG$-module $M$ is relative projective module for $H$ exactly when there is $\alpha \in {\mathrm{End}}_{RH}(M)$ satisfying

If $[G:H]$ is invertible in $R$, every $RG$-module is relatively H-projective. An $RG$-module is then projective exactly when its restriction to $RH$ is projective.

A vertex of an indecomposable $kG$-module is a subgroup minimal among those for which the module is relatively projective. Vertices exist, form one conjugacy class, and are p-groups when $k$ has characteristic $p$. The vertices of the trivial module are the Sylow p-subgroups.

A group algebra $kG$ has finite representation type when it has only finitely many isomorphism classes of finite-dimensional indecomposable modules.

If $k$ has characteristic $p$, then $kG$ has finite representation type exactly when a Sylow p-subgroup of $G$ is cyclic. Restriction and induction reduce the property to the Sylow subgroup; a cyclic p-group has the finitely many indecomposables $k[u]/(u^r)$, while a noncyclic p-group has a quotient $C_p\times C_p$ and hence infinitely many indecomposables.

A simple projective module over a split modular group algebra lies in a block of defect zero. It lifts uniquely to an ordinary irreducible character whose degree is divisible by the full p-part of the group order.

A block of a finite group algebra is an indecomposable two-sided ideal $RGe$ determined by a primitive central idempotent $e$. An $RG$-module $M$ lies in this block when $eM=M$.

A defect group of a block idempotent $e$ is a maximal p-subgroup $D$ for which the Brauer image ${\mathrm{Br}}_D(e)$ is nonzero. Equivalently, the vertices of the block algebra as an $R[G\times G]$-module are the diagonal subgroups $\Delta D$ for the defect groups $D$. All defect groups of a block are conjugate.

For a p-subgroup $D\le G$, let $D$ act on $kG$ by conjugation. The Brauer morphism is the algebra homomorphism

Let $D$ be a p-subgroup of $G$, put $N=N_G(D)$, and let $k$ have characteristic $p$. The Brauer morphism intertwines the two relative traces:Indeed, $D$ acts on $G/D$ by left multiplication. A coset $gD$ is fixed exactly when $g\in N$, and every other orbit has size divisible by $p$. After applying ${\mathrm{Br}}_D$, the summands belonging to one such orbit are equal, so every nonfixed orbit contributes zero in characteristic $p$; the fixed cosets give the trace from $D$ to $N$.

Let $B$ be a block of $kG$ with defect group $D$ and put $N=N_G(D)$. Its Brauer correspondent is the unique block $b$ of $kN$ with defect group $D$ selected by the nonzero Brauer image of the block idempotent of $B$.

Brauer's first main theorem gives a bijection between the blocks of $kG$ with defect group $D$ and the blocks of $k{N}_G(D)$ with defect group $D$. Corresponding blocks are related by their images under the Brauer morphism.

Over a splitting field of characteristic five, $A_5$ has a principal block of defect $C_5$ containing the ordinary characters of degrees $1,3,3,4$, and one defect-zero block containing the ordinary character of degree $5$. If $P$ is a Sylow 5-subgroup, then $N_{A_5}(P)\cong D_{10}$ and its unique 5-block is the Brauer correspondent of the principal block of $A_5$.

The decomposition matrix records the multiplicities of simple modular representations in reductions of ordinary representations. If its rows are indexed by ordinary irreducible characters and its columns by irreducible Brauer characters, its entry $d_{\chi \varphi }$ is the decomposition number of $\varphi$ in the reduction of $\chi$.

The Cartan matrix records composition-factor multiplicities in the projective indecomposable modules. For a split modular group algebra, it iswhere $D$ is the decomposition matrix.

The simple modules have dimensions $1,3,3,8$. The eight-dimensional simple is projective and forms a defect-zero block; the other three simples belong to the principal block. The decomposition and Cartan matrices are computed by restricting the six ordinary irreducible characters to the four odd-order conjugacy classes.