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.

A p-local module here is a finite direct sum of induced representations from normalizers $N_G(Q)$ of nontrivial p-subgroups $Q$. This usage includes induction from $G$ itself when $G$ normalizes such a subgroup. Green correspondence and induction on vertex order express every finite module, or lattice over a complete valuation ring, as a difference of p-local modules after adjoining projective modules.

Here $\mathcal{O}$ is a complete discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p$. The system is splitting for the finite group and its subgroups when their simple representations over $K$ and $k$ are absolutely irreducible. It permits ordinary and modular representation theory to be compared by reduction of lattices modulo the maximal ideal.

For $P$ cyclic of order $q=p^n$ in characteristic $p$, its group algebra is $k[t]/(t^q)$. The nilpotent Jordan normal form classifies its indecomposable finite-dimensional modules as $V_i$, $1\le i\le q$. A module homomorphism $V_i\to V_j$ is determined by an element annihilated by $t^i$, so its dimension is $\min (i,j)$.

The modular representation ring is the Grothendieck group of finite-dimensional group representations, with relations $[V]=[U]+[W]$ for every short exact sequence $0\to U\to V\to W\to 0$, and multiplication $[V][W]=[V{\otimes }_{k}W]$ using the tensor product of group representations. The Jordan–Hölder theorem makes the classes of simple modules a free integral basis. Tensoring over a field is exact, so this multiplication is well defined. When $k$ is a splitting field for finite group representations, Brauer characters identify $\mathbb{C}{\otimes }_{\mathbb{Z}}{R}_k(G)$ with the algebra of complex class functions on p-regular elements. The splitting hypothesis is essential: ${\mathbb{F}}_2{C}_3\cong {\mathbb{F}}_2\times {\mathbb{F}}_4$ has two simple modules, although $C_3$ has three $2$-regular conjugacy classes.

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.

For a p-modular system $(K,\mathcal{O},k)$, an integral form of a finite-dimensional $KG$-module $V$ is a $G$-stable finite free $\mathcal{O}$-submodule $W\subset V$ with $K{\otimes }_{\mathcal{O}}W\cong V$. Starting with a basis lattice $L$, the sum ${\sum }_{g\in G}gL$ supplies such a form: it is finitely generated and torsion-free, hence free over the discrete valuation ring $\mathcal{O}$. Reduction gives the $kG$-module $W/\pi W$, where $\pi$ is a uniformizer.

Let $W,W^{\prime }$ be choices of integral form of a group representation over a complete p-modular system. Then $I={\mathrm{Hom}}_{\mathcal{O}G}(W,W^{\prime })$ is finite free, and $K{\otimes }_{\mathcal{O}}I={\mathrm{Hom}}_{KG}(K\otimes W,K\otimes W^{\prime })$: clearing denominators proves the spanning assertion. Also ${\mathrm{Hom}}_{\mathcal{O}G}(W,\pi W^{\prime })=\pi I$. If $W$ is a projective module, applying the Hom functor to $0\to \pi W^{\prime }\to W^{\prime }\to W^{\prime }/\pi W^{\prime }\to 0$ yieldsConsequently these ordinary and modular Hom spaces have the same dimension of a vector space. Projectivity is essential: for $G=C_2$ over ${\mathbb{Z}}_2$, the trivial and sign lattices have zero Hom between them, whereas their reductions in characteristic $2$ coincide and have a one-dimensional Hom space.

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.

Over a splitting field for finite group representations of characteristic $p$, the irreducible Brauer characters form a complex basis of the class functions on the p-regular elements. Independence is the character form of the Brauer–Nesbitt theorem. To obtain spanning, extend such a class function by zero on the p-singular classes. Ordinary irreducible characters form a basis of all class functions by character orthogonality. Restricting them to p-regular elements yields Brauer characters of reductions of an integral form of a group representation in a compatible splitting p-modular system; if needed, first extend scalars, which does not change the simple-module list under the splitting hypothesis. Each restriction is a nonnegative integral sum of simple Brauer characters by exact-sequence additivity and the Jordan–Hölder theorem. Thus these restrictions span, proving the assertion. Consequently the number of simple modules equals the number of p-regular conjugacy classes, and evaluation identifies the complexified modular representation ring with the product of one copy of $\mathbb{C}$ for each such class.

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-dimensional symmetric algebra over any field, an indecomposable projective module $P=Ae$ has isomorphic simple head of a module and socle. A symmetrizing functional pairs $eA$ with $Ae$ nondegenerately, identifying $Ae$ with the left dual module $D(eA)$. Taking its socle gives $D(eA/eAJ(A))$. This dual of the right simple top and the left simple top $Ae/J(A)e$ belong to the same matrix block of the semisimple algebra $A/J(A)$, so they are isomorphic. The result does not require an algebraically closed ground field.

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.

Every noncyclic finite finite p-group $P$ has a quotient $C_p\times C_p$. Over any field of characteristic $p$, define a square-zero representation with basis $v_0,\dots ,v_r,w_1,\dots ,w_r$ by $x{v}_i=w_i$ for $1\le i\le r$, $x{v}_0=0$, $y{v}_i=w_{i+1}$ for $0\le i<r$, $y{v}_r=0$, and $xW=yW=0$. Here $x=g-1$, $y=h-1$, and $W$ is the span of the $w_i$. An endomorphism induces the same scalar on the top and on $W$; the remaining maps from the top to $W$ form a square-zero ideal. Thus the endomorphism ring is local and the module is indecomposable. Inflating to $P$ and choosing $p\nmid 2r+1$ gives infinitely many distinct dimensions with vertex $P$: the trace of a relative-trace identity from any proper subgroup would force dimension zero in the field. This construction works over finite fields too.

Let $N_G(D)\le H\le G$. Set $\mathcal{X}=\{D\cap {}^{g}D:g\notin H\}$ and $\mathcal{Y}=\{H\cap {}^{g}D:g\notin H\}$. Green correspondence pairs indecomposable modules with vertex $D$ for $G$ and $H$: restriction has a unique vertex-$D$ summand, with the other summands relatively projective for $\mathcal{Y}$; induction has a unique vertex-$D$ summand, with the other summands relatively projective for $\mathcal{X}$. The two assignments are inverse and preserve sources. Its more general form applies to vertices contained in $D$ but not conjugate into any member of $\mathcal{X}$.

Let $N_G(D)\le H\le G$ and let $U$ be indecomposable with vertex $D$ over $H$. The Mackey restriction formula writes ${\mathrm{Res}}_H^G{\mathrm{Ind}}_H^{G}U=U\oplus T$, where the vertices in $T$ lie in $H\cap {}^{g}D$ for $g\notin H$. None of these subgroups contains an $H$-conjugate of $D$: equality would put $g$ in $N_G(D)H=H$. Every summand of the induction with a $G$-conjugate of vertex $D$ contributes a vertex-$D$ summand on restriction. Krull-Schmidt decomposition therefore forces exactly one such summand, with multiplicity one. Choosing sources and using transitivity of induction proves the inverse correspondence and preservation of sources.

The Green correspondent is the unique indecomposable summand of restriction or induction having the selected vertex of an indecomposable module, in the setting of Green correspondence. The two correspondence maps are inverse and preserve sources of an indecomposable module up to the appropriate normalizer conjugacy.

If $N_G(D)\le H\le G$, Brauer correspondence pairs the blocks of a group algebra of $RG$ and $RH$ having defect group of a block $D$. This is Green correspondence for their bimodules, with vertex of an indecomposable module $\Delta D$ in $G\times G$. The normalizer of $\Delta D$ lies in $H\times H$, and the corresponding restriction and induction have unique summands with that vertex.

For an indecomposable module $M$ with vertex of an indecomposable module $D$, a source is an indecomposable $RD$-module $S$ which is a direct summand of ${\mathrm{Res}}_D^{G}M$ and for which $M$ is a direct summand of ${\mathrm{Ind}}_D^{G}S$. It has vertex $D$ as a $D$-module. Sources at a fixed vertex are conjugate by its normalizer.

Over a modular field or a complete discrete valuation coefficient ring, an indecomposable module has trivial source when a source is the rank-one trivial module of its vertex. These are precisely the indecomposable direct summands of permutation modules. A general trivial source module is a direct sum of these indecomposables.

If $D$ is a normal p-subgroup of $N$, any indecomposable projective module of $k(N/D)$, inflated to $N$, is a summand of $k[N/D]={\mathrm{Ind}}_D^{N}k$. It has trivial source and vertex $D$. Its Brauer quotient of a module at $D$ is the entire module, since $D$ acts trivially and all proper-subgroup relative traces multiply by an index divisible by $p$. This prevents its vertex from being smaller than $D$.

For coset permutation modules, Mackey restriction formula and Frobenius reciprocity identify the homomorphism space with a free module having one orbit-sum basis vector per double coset. The same basis works over the residue field, so reduction of homomorphisms is surjective. Applied to endomorphisms of a permutation module, idempotent refinement theorem lifts each modular direct summand to a trivial source lattice. Lifting inverse isomorphisms between reductions shows uniqueness among trivial source lattices.

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

For $P=C_p\times C_p$, put $x=g-1$, $y=h-1$. On a basis $v,w$, prescribe $xv=w$, $xw=0$, $yv=\lambda w$, $yw=0$. These are valid modules because every product of two radical operators is zero. Each is indecomposable, and an isomorphism preserves the equality $y=\lambda x$, hence preserves $\lambda$. An infinite field therefore gives infinitely many isomorphism classes.

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 $eM=M$.

For shapes of the same size, equal residue-content multiplicities are equivalent to equal cores of a partition at modulus $p$. Removing a rim $p$-hook subtracts one of each residue. With the number of beta numbers divisible by $p$, the packed-runner charges are $q_r=c_r-c_{r+1}$, so the multiplicities recover the core. Symmetric polynomials in Young–Jucys–Murphy elements act by their evaluations on these residues, and generate the modular center. Equality of its characters is precisely the block-equivalence criterion for simple modules.

At fixed degree $n$ in positive characteristic $p$, two Specht modules belong to the same block of a group algebra exactly when their labels have the same core of a partition at modulus $p$. The simple labels in that block are precisely the regular partitions with that core. The theorem follows by combining the residue-content criterion for symmetric-group blocks with the Jucys–Murphy description of the center of a symmetric-group algebra.

The center of an indecomposable finite-dimensional block of a group algebra is a commutative Artinian local ring. Its quotient by its Jacobson radical is a field, giving the block's central character. Over a splitting field for finite group representations the residue field is the coefficient field; without this assumption one retains the residue field rather than pretending every central character is scalar-valued over $k$.

Let $N◃G$, let $b$ be a block of a group algebra of $kN$, and let $I$ be its inertia subgroup. A block of $kG$ covering $b$ is a full matrix algebra of degree $[G:I]$ over its corresponding block of $kI$. When $I=N$, the corner is just $kNb$. Let $e_B$ be the global block idempotent and put $c=e_{B}b$. With coset representatives $g_i$, the elements $g_{i}c{g}_j^{-1}$ are matrix units in that global block because different conjugates of $b$ are orthogonal. When $I=N$, one has $c=b$.

The principal block of a group algebra is the unique block of a group algebra containing the trivial representation. Its primitive central idempotent is the unique one with augmentation $1$. Its defect groups of a block are the Sylow subgroups for the residue characteristic.

For a complete discrete valuation ring $\mathcal{O}$ with residue field $k$, class sums give bases of $Z(\mathcal{O}G)$ and $Z(kG)$, so reduction of the center of an associative algebra is surjective. Idempotent lifting in this complete commutative algebra gives a bijection of primitive central idempotents, hence of blocks of a group algebra. Corresponding blocks of a group algebra have the same defect groups of a block: relative traces reduce, and a lifted trace whose reduction is the block identity is a unit in its complete block center.

Let $Z=⟨z⟩\le Z(G)$ have order $p$ and set $t=z-1$. The nilpotent ideal $tkG$ is generated by a central element. If an idempotent lifts a central idempotent of the quotient, its off-diagonal corner satisfies $eA(1-e)\subseteq teA(1-e)$ and therefore is zero. Thus central block idempotents lift centrally and uniquely. A principal indecomposable module is free over $kZ$, and the $p$ factors of its $t$-filtration are all its reduction modulo $t$. Consequently, under the natural indexing of simple modules, $C_G=p{C}_{G/Z}$.

Over a field of characteristic three, the group algebra $A=k{S}_3$ has a single block of an Artinian algebra. Put $r=(123)$, $s=(12)$ and $a=r-1$. Then $a^3=0$, $sas=-a+a^2$, and $J(A)=aA$: the latter is a nilpotent ideal, with semisimple ring quotient $k{C}_2$. The center of an associative algebra is $k1\oplus k{a}^2\oplus k{a}^{2}s$, whose last two summands form a square-zero ideal. A central idempotent $d1+n$ satisfies $d^2=d$ and $(2d-1)n=0$, hence is zero or one. Thus no nontrivial central block decomposition exists.

With $s=(12)$, the idempotents $e_{\pm }=(1\pm s)/2$ split the right regular module into two three-dimensional projective modules. Their tops are respectively the trivial representation and sign representation. The nilpotent ideal $J(k{S}_3)$ ensures that any nonzero direct summand has nonzero top, so the one-dimensional tops make these projective modules indecomposable. Their successive radical series of a module factors are trivial, sign, trivial and sign, trivial, sign. This decomposition concerns right modules; the block of S3 in characteristic three remains a single algebra.

Over a splitting field for finite group representations of characteristic $2$, put $a=(123)$. The block of a group algebra idempotents of $k{S}_3$ are $b_0=1+a+a^2$ and $b_1=a+a^2$. Their block algebras are $b_{0}k{S}_3\cong k{C}_2$ and $b_{1}k{S}_3\cong M_2(k)$, with a defect group of a block given by $C_2$ and $1$, respectively. The first is a local ring; for the second, the representation $a\mapsto \mathrm{diag}(\omega ,{\omega }^2)$, $(12)\mapsto \left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ generates the full matrix algebra. The Cartan matrix of a group algebra is $\mathrm{diag}(2,1)$, and the ordinary trivial, sign, and two-dimensional characters give decomposition matrix $\left(\begin{array}{cc}1& 0\\ 1& 0\\ 0& 1\end{array}\right)$.

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 block idempotent $b$ with defect $D$, the Brauer quotient of a module of its modular block bimodule at $\Delta D$ is nonzero: on the fixed basis $C_G(D)$, the block projection acts by ${\mathrm{Br}}_D(b)$. Restriction to $D\times D$ splits into transitive permutation modules; their stabilizers have order at most $|D|$. A nonzero quotient at $\Delta D$ therefore forces a summand with stabilizer exactly $\Delta D$, which is the regular $kD$ bimodule. Tensoring on the right with any $kD$-module $U$ gives $U\mid {\mathrm{Res}}_D^G(b{\mathrm{Ind}}_D^{G}U)$. Consequently finite representation type of the modular block forces finite representation type of $kD$.

Let $P$ be any Sylow subgroup containing a defect group of a block $D$. Regard the block of a group algebra as a permutation module summand for $G\times G$, acting by left and right multiplication. Its vertex is $\Delta D$. Restricting to $P\times P$ retains a summand with this vertex. Transitive permutation modules for a finite p-group are indecomposable, with their point stabilizers as vertices. Thus some orbit in $G$ has stabilizer $\Delta D$ at a suitably chosen point $c$. This means $c$ centralizes $D$ and $P\cap {}^{c}P=D$. Choosing $P$ to contain a Sylow subgroup of $N_G(D)$ shows that $D=O_p(N_G(D))$; every normal p-subgroup of $G$ is contained in both Sylow factors and hence in $D$.

For a block of a group algebra with identity $e$ and defect group of a block $D$, the trace criterion for defect groups writes $e={\mathrm{Tr}}_D^G(a)$ with $a\in (RG{)}^D$. On any module in the block, multiplication by $a$ is $D$-linear and its relative trace is the identity. The D. Higman criterion therefore makes the module relatively $D$-projective.

Let $k$ be an algebraically closed field and let a block of a group algebra $B$ have a central defect group of a block $D$. Its quotient by $J(kD)B$ is a defect-zero block, hence a full matrix algebra over $k$. The algebra $B$ is free over the central local ring $kD$. Lift a primitive idempotent from this matrix quotient; its corner is free of rank one over $kD$, and is therefore $kD$. Lifting matrix units gives $B\cong {\mathrm{Mat}}_r(kD)$.

For a maximal Brauer pair $(D,b_D)$ of a block of a group algebra, the inertial quotient is the displayed quotient of its pair stabilizer by $D{C}_G(D)$. It has order coprime to $p$, and its order is the inertial index of a block. For abelian $D$, conjugation embeds it in $\mathrm{Aut}(D)$.

Choose a maximal Brauer pair $(D,b_D)$ for the block. Its inertial quotient is $N_G(D,b_D)/(D{C}_G(D))$, and its order is the inertial index. For cyclic $D$ of order $q$, this is a prime-to-$p$ divisor of $p-1$. The number of edges of the block's Brauer tree is this index, and the exceptional multiplicity is $(q-1)/e(B)$.

The defect of a block of a group algebra is the integer $d$ for which a defect group of a block has order $p^d$. Conjugacy of defect groups of a block makes this independent of the choice. A block of a group algebra has defect zero when its defect group of a block is trivial.

For a block idempotent $e$, an $e$-Brauer pair consists of a $p$-subgroup $P$ and a block idempotent $b_P$ of $k{C}_G(P)$ for which $b_P{\mathrm{Br}}_P(e)\ne 0$. The subgroup of a maximal pair is a defect group. A maximal pair's stabilizer defines the inertial quotient of the block.

A block has finite representation type exactly when its defect group is cyclic. In the forward implication, the block bimodule restricted to its defect group on both sides contains the regular defect-group bimodule as a summand; tensoring with arbitrary defect-group modules transfers finiteness to the defect group. Conversely, vertices are contained in the defect group and a cyclic $p$-group has only finitely many indecomposable modules, so there are only finitely many possible induced sources and their summands.

The basic algebra of a split block with cyclic defect is a Brauer tree algebra: its simple modules are indexed by the edges of a tree, with a distinguished vertex carrying the exceptional multiplicity. For a one-edge tree of multiplicity $m$, its basic algebra is $k[t]/(t^{m+1})$. Its ideals form the chain of powers of $t$, so its unique indecomposable projective is uniserial of length $m+1$.

Let $i$ be an edge at $v$. Kill the other radical branch of its indecomposable projective module. The resulting uniserial module has length $m_v\mathrm{val}(v)$ and its factors follow the cyclic edge order beginning at $i$. Quotienting its radical powers gives all shorter such modules. The edge succession uniquely fixes the path, and its nonzero arrow maps can be rescaled to one, giving uniqueness up to isomorphism. At an exceptional leaf this reduces to a nilpotent single Jordan chain; the other leaf loop, when present, is a socle path and vanishes in these shorter quotients.

Use a finite tree with cyclically ordered incident edges at each vertex, and multiplicities one except possibly at one vertex. Quiver vertices are tree edges; arrows follow each cyclic successor order, including loops at vertices of valency one. For an edge $i$ with endpoints $v,w$, write $C_{i,v}$ for the cycle around $v$. Impose $C_{i,v}^{m_v}=C_{i,w}^{m_w}$, $C_{i,v}^{m_v}{\alpha }_{i,v}=0$, and zero for any path that switches from one vertex-cycle to the other. The quotient of this path algebra is the basic Brauer tree algebra; redundant valency-one loops can be eliminated. Its indecomposable projective at $i$ has simple top and socle labelled by $i$, with two uniserial radical branches of lengths $m_v\mathrm{val}(v)$ and $m_w\mathrm{val}(w)$ meeting in that socle.

For a split block of a group algebra with cyclic defect group of a block of order $q=p^n$ and inertial index of a block $e$, its Brauer tree algebra has $e$ simple modules and exceptional multiplicity $(q-1)/e$. Its stable Auslander–Reiten quiver is $\mathbb{Z}{A}_{q-1}/⟨{\tau }^e⟩$, giving $e(q-1)$ nonprojective indecomposables. Adding the $e$ indecomposable projectives gives $eq$ indecomposables in total.

In characteristic $2$, a Sylow subgroup $P$ of $A_5$ is a Klein four-group, and its normalizer is $A_4$. Since $k{A}_4$ has one block, Brauer first main theorem gives a unique block of $k{A}_5$ with defect $P$. No order-two subgroup $D$ can be a defect group of a block: both $C_{A_5}(D)$ and $C_{A_5}(P)$ equal $P$, so the Brauer morphisms of a central block idempotent at $D$ and at $P$ are equal. A nonzero image is $1$, since a group algebra of a p-group in characteristic p is local, and consequently $D$ is not maximal among subgroups with nonzero Brauer image.

For a $p$-subgroup $D$ of a finite group $G$, set $I_D={\mathrm{Tr}}_D^G((kG{)}^D)$, a transfer ideal of conjugation-fixed elements in $Z(kG)$. A block of a group algebra idempotent $b$ lies in $I_D$ if and only if a defect group of a block of $b$ is conjugate to a subgroup of $D$. In particular, defect groups can equivalently be defined as minimal subgroups $D$ with $b\in I_D$.

Here is a proof using the Brauer morphism. A subgroup of smallest order with $b\in I_D$ exists, since a Sylow subgroup $P$ gives $b={\mathrm{Tr}}_P^G(b/[G:P])$. Write $b={\mathrm{Tr}}_D^G(a)$ and replace $a$ by $ba$. If ${\mathrm{Br}}_D(b)=0$, then $ba$ belongs to the kernel of the Brauer morphism, which is the sum of proper-subgroup relative traces. Hence $b\in {\sum }_{Q<D}{I}_Q$. The algebra $bZ(kG)$ is a commutative Artinian ring and a local ring, so multiplying this equality by $b$ forces some ideal $b{I}_Q$ to contain its identity $b$. This contradicts minimality. On the other hand, ${\mathrm{Br}}_E(I_D)=0$ unless $E$ is conjugate into $D$: the $E$-action on $G/D$ has no fixed coset otherwise, and nonfixed orbit sizes vanish in characteristic $p$. Thus these minimal subgroups are precisely the maximal subgroups with nonzero Brauer image. They are all conjugate. Trace transitivity proves the stated criterion for larger $D$.

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

For a normal p-subgroup $P◃G$, every central idempotent $e$ of $kG$ satisfies $e={\mathrm{Br}}_P(e)\in k{C}_G(P)$. Indeed a nonfixed conjugation orbit sum maps to zero in $k(G/P)$, so $e-{\mathrm{Br}}_P(e)$ belongs to the nilpotent ideal $J(kP)kG$. Normality makes the Brauer image a central idempotent of $kG$, and two central idempotents congruent modulo a nilpotent ideal coincide. Consequently the blocks of a group algebra of $kG$ correspond to $G$-orbits of blocks of $k{C}_G(P)$.

For a $p$-subgroup $P$, the Brauer quotient removes all fixed vectors produced by proper-subgroup relative traces. On a permutation module it has a basis of the permutation-basis elements fixed by $P$: nonsingleton orbit sums are traces and disappear. The construction commutes with direct sums and idempotent projections. For an indecomposable trivial source module, it is nonzero exactly when a vertex contains a conjugate of $P$.

For $N=N_G(D)$ and $C=C_G(D)$, the Brauer morphism restricts to a surjectionThis follows from Brauer morphism and relative trace and the fact that $kC$ is contained in $(kG{)}^D$. The target is an ideal in the commutative algebra $(kC{)}^N$. Applying primitive idempotents under a surjection of commutative Artinian ideals and the trace criterion for defect groups identifies block idempotents with defect group $D$ with the primitive idempotents of $J_D$. Applying the same argument to $N$ gives Brauer first main theorem. The whole kernel need not be nilpotent, since blocks with smaller defect can be killed.

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 $B$.

The Brauer correspondent of a block of a group algebra with defect group of a block $D$ in $G$ is the corresponding block of $N_G(D)$ furnished by Brauer first main theorem. The more general induced block $b^G$ is defined by block induction when the subgroup contains the centralizer of a defect group of $b$.

If a block of a group algebra $b$ of $kH$ has defect group of a block $D$ and $D{C}_G(D)\le H\le G$, then its block induction $b^G$ is the principal block if and only if $b$ is the principal block. Equivalently, the Brauer morphism of the principal block idempotent at any p-subgroup $Q$ is the principal block idempotent of $k{C}_G(Q)$.

Let $b$ be a block of a group algebra of $kH$ with defect group of a block $D$, where $D{C}_G(D)\le H\le G$. Apply Brauer first main theorem inside $H$ to obtain its correspondent in $N_H(D)$. Composing that correspondent's central character of a block with the Brauer morphism from $Z(kG)$ selects a unique block of a group algebra $b^G$. It is the unique block of a group algebra whose restriction to $H\times H$ contains the block bimodule $b$ as a direct summand. Its defect group of a block can be larger than $D$ when $H$ does not contain $N_G(D)$.

The centralizer condition defining block induction refers to a defect group of a block of the actual source block. Containment of the centralizer of an unrelated p-subgroup does not suffice. For example, in characteristic $3$ take $G=A_5$, $H=A_4$ and $D=C_3$. Then $D{C}_G(D)\le H$, but the unique defect-zero block of $k{A}_4$ occurs on bimodule restriction of both three-dimensional defect-zero blocks of $k{A}_5$. The two ordinary degree-three characters have identical restriction to $A_4$, namely its degree-three character, and defect-zero reduction gives the same simple projective module. Thus the unique-block characterization fails if $D$ is not the defect group of the source block.

For a central idempotent $e$, a module $M=eM$, and $C_G(D)\le K\le N_G(D)$, put $b={\mathrm{Br}}_D(e)$. Then $M{\downarrow }_K=bM\oplus (1-b)M$, and every indecomposable summand of the second term has vertex not containing $D$. Indeed $e-b$ is a sum of $K$-orbit sums of group elements not centralizing $D$. Each is a relative trace from a centralizer not containing $D$. Applying these traces to the module and using the local endomorphism ring of an indecomposable summand proves relative projectivity for one such subgroup.

For an indecomposable $kK$-module $V$ with vertex $D$ and ${\mathrm{Br}}_D(e)V=V$, every indecomposable summand of $(1-e){\mathrm{Ind}}_K^{G}V$ has vertex conjugate into $D\cap {}^{g}D$ for some $g\notin N_G(D)$. The refinement keeps the intersection subgroups from Green correspondence, rather than asserting only that the remaining vertices are smaller than $D$.

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 for finite group representations, blocks of a group algebra with defect group of a block $D$ correspond to $N_G(D)$-orbits of blocks $b_D$ of $k(D{C}_G(D))$ with defect $D$ whose stabilizer quotient $N_G(D,b_D)/(D{C}_G(D))$ has order coprime to $p$. Equivalently one uses the defect-zero block of $k(D{C}_G(D)/D)$. This quotient is the inertial quotient of a block.

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 is a specific invariant of modular representation theory.

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 $\chi$.

Rows of the symmetric-group decomposition matrix are all ordinary shapes, columns are regular partitions, and $d_{\mu \lambda }=[S^{\mu }:D^{\lambda }]$. A nonzero column antisymmetrizer of shape $\lambda$ on $M^{\mu }$ forces dominance: the first $r$ rows contain at most ${\sum }_j\min (r,{\lambda }_j^{\prime })$ column entries. For a regular shape its diagonal simple appears once, giving a rectangular triangular matrix and a unitriangular regular-row submatrix.

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.