Block of a group algebra

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.