Remarks on the extended Brauer quotient

TL;DR

This paper discusses the graded structure of the extended Brauer quotient, an interior G-algebra, highlighting its algebraic properties and implications.

Contribution

It introduces the graded structure of the extended Brauer quotient as an interior G-algebra, providing new insights into its algebraic framework.

Findings

Identification of the graded structure of the extended Brauer quotient

Insights into the algebraic properties of interior G-algebras

Potential implications for modular representation theory

Abstract

We point out the graded structure of the extended Brauer quotient an interior $G$-algebra.

Equations

$${\Delta }_{\phi }(P)=\{(u,\phi (u))\mid u\in P\}.$$

$$A^{{\Delta }_{\phi }(P)}=\{a\in A\mid ua=a\phi (u)\text{mbox}foranyu\in P\}.$$

$${\mathrm{Tr}}_{{\Delta }_{\phi }(Q)}^{{\Delta }_{\phi }(P)}(c)=\sum _{u\in [P/Q]}{u}^{-1}c\phi (u),$$

$$A({\Delta }_{\phi }(P))=A^{{\Delta }_{\phi }(P)}/\sum _{Q<P}{A}_{{\Delta }_{\phi }(Q)}^{{\Delta }_{\phi }(P)},$$

$${\mathrm{Br}}_{{\Delta }_{\phi }(P)}:A^{{\Delta }_{\phi }(P)}\to A({\Delta }_{\phi }(P)).$$

$${\bar{N}}_A^K(P):=\underset{\phi \in K}{⨁}{A}^{{\Delta }_{\phi }(P)}/\underset{\phi \in K}{⨁}\sum _{Q<P}{A}_{{\Delta }_{\phi }(Q)}^{{\Delta }_{\phi }(P)}\simeq \underset{\phi \in K}{⨁}A({\Delta }_{\phi }(P)).$$

$$N_G(P)\to \mathrm{Aut}(P)\mathrm{and}{N}_G(P)/C_G(P)\to \mathrm{Aut}(P),$$

$${\bar{N}}_A^K(P)\simeq A(P){\otimes }_{k{C}_G(P)}k{N}_G(P)$$

$$A(P){\otimes }_{k{C}_G(P)}k{N}_G(P)\to {\bar{N}}_A^K(P),\bar{a}\otimes x\mapsto \overline{ax},$$

$$\tilde{A}:=A{\otimes }_P\mathcal{O}(P⋊\mathrm{Aut}(P)).$$

$$\tilde{A}\simeq A{\otimes }_{\mathcal{O}}\mathcal{O}\mathrm{Aut}(P)$$

$$u(a\otimes \varphi )v=u\cdot a\cdot \varphi (v)\otimes \varphi ,$$

$$(a\otimes \varphi {)}^u=u^{-1}\cdot a\cdot \varphi (u)\otimes \varphi ,$$

$$\tilde{A}(P)\simeq {\bar{N}}_A^{\mathrm{Aut}(P)}(P)$$

$$\tilde{A}(P)=\underset{\phi \in \mathrm{Aut}(P)}{⨁}(A\otimes (1,\phi ))(P).$$

$$u^{-1}\cdot a\otimes (1,\phi )\cdot u=u^{-1}\cdot a\otimes (1,\phi )(u,1)=u^{-1}a\phi (u)\otimes (1,\phi )=a\otimes (1,\phi ).$$

$$(A\otimes (1,\phi ))(P)\to {\bar{N}}_A^{\phi }(P),\overline{a\otimes (1,\phi )}\mapsto \overline{a},$$

$$(A{\otimes }_{\mathcal{O}Q}\mathcal{O}[Q⋊K])(Q)\simeq {\bar{N}}_A^K(Q)$$

$$A:=B{\otimes }_{\mathcal{O}H}\mathcal{O}G$$

$$a={\mathrm{Tr}}_{{\Delta }_{\phi }(Q)}^{{\Delta }_{\phi }(P)}(b_z\otimes z),$$

$${\mathrm{Aut}}_{\bar{G}}(P)=\{\phi \in \mathrm{Aut}(P)\mid \overline{\phi (u)}=\overline{u^g}\text{mbox}forsome\bar{g}\in \bar{G}\text{mbox}andforanyu\in P\}$$

$${\mathrm{Aut}}_{\bar{1}}(P)=\{\phi \in \mathrm{Aut}(P)\mid \overline{\phi (u)}=\bar{u}\text{mbox}foranyu\in P\}.$$

$$K:={\mathrm{Aut}}_{\bar{G}}(P),K_1:={\mathrm{Aut}}_{\bar{1}}(P),$$

$${\mathrm{Aut}}_{\bar{G}}(P)/{\mathrm{Aut}}_{\bar{1}}(P)\simeq N_{\bar{G}}^K(\bar{P})/C_{\bar{G}}(\bar{P})\simeq U/U^{\prime }.$$

$$\overline{({\phi }^{-1}\circ {\phi }_1\circ \phi )(u)}=\overline{({\phi }_1\circ \phi )(u{)}^{g^{-1}}}=\overline{\phi (u{)}^{g^{-1}}}=\bar{u}.$$

$${\mathrm{Aut}}_{\bar{G}}(P)\ni \phi \mapsto \bar{g}\in N_{\bar{G}}(\bar{P})$$

$${\bar{N}}_A^{K_1}(P)=\underset{g^{\prime }\in U^{\prime }/H}{⨁}{\bar{N}}_{B\otimes g^{\prime }}^{K_1}(P),$$

$${\bar{N}}_A^K(P{)}_{\tilde{g}}=\underset{{\phi }_{\bar{z}}\in \tilde{\phi };\bar{z}\in \tilde{g}}{⨁}(B\otimes z)({\Delta }_{{\phi }_{\bar{z}}}(P)),$$

$${\bar{N}}_A^{\mathrm{Aut}(P)}(P)$$

$$=\left(\underset{{\bar{g}}^{\prime }\in U^{\prime }/H}{⨁}{\bar{N}}_{B\otimes g^{\prime }}^{K_1}(P)\right)$$

$$\oplus \underset{\tilde{\phi }\in K/K_1}{⨁}\left(\underset{{\phi }_{\bar{z}}\in \tilde{\phi };\bar{z}\in \tilde{g}}{⨁}(B\otimes z)({\Delta }_{{\phi }_{\bar{z}}}(P))\right),$$

Full text

1 Introduction

Let $G$ be a group, and let $A$ be a $G$-algebra over a complete discrete valuation ring $\mathcal{O}$ with residue field $k$ of characteristic $p>0$. The well-known Brauer quotient $A(P)$ with respect to a $p$-subgroup $P$ of $G$ (introduced by M. Broué and L. Puig, see [8, §11]) is an $N_{G}(P)$-algebra. If moreover, $A$ is $G$-interior (that is, $A$ is endowed with a unitary algebra homomorphism $\mathcal{O}G\to A$), then $A(P)$ becomes a $C_{G}(P)$-interior $N_{G}(P)$-algebra. This means that one may construct, as in [5, Chapter 9], the $N_{G}(P)/C_{G}(P)$-graded $N_{G}(P)$-interior algebra $A(P)\otimes_{C_{G}(P)}N_{G}(P)$, so $A(P)$ is extended by automorphisms of $P$ given by conjugation with elements of $G$.

L. Puig and Y. Zhou [6] extended $A(P)$ by all automorphisms of $P$, obtaining the so called extended Brauer quotient $\bar{N}_{A}^{\operatorname{Aut}(P)}(P)$ as an $N_{G}(P)$-interior $k$-algebra. The interiority assumption is necessary, because the main feature used is the $\mathcal{O}(P\times P)$-module structure of $A$. This construction was further generalized by T. Coconeţ and C.-C. Todea [3] to the case of $H$-interior $G$-algebras, where $H$ is a normal subgroup of $G$.

Our aim here is to unify and generalize these constructions, by introducing an extended Brauer quotient of a group graded algebra. The main ingredients of our construction are a $\bar{G}$-graded algebra $A$, a group homomorphism $P\to\bar{G}$ (which induces a $\bar{G}$-grading on the group algebra $\mathcal{O}P$), and a homomorphism $\mathcal{O}P\to A$ of $\bar{G}$-graded algebras.

In Section 2 below we recall the Puig and Zhou definition of $\bar{N}_{A}^{\operatorname{Aut}(P)}(P)$, pointing out its $\operatorname{Aut}(P)$-graded algebra structure. Our alternative construction in Section 3 is based on the easy observation that if $A$ is a $\bar{G}$-graded $P$-algebra with identity component $B$ such that the action of $P$ on $\bar{G}$ is trivial, then the Brauer quotient $A(P)$ inherits the $\bar{G}$-grading such that the identity component of $A(P)$ is $B(P)$. Here we apply the classical Brauer quotient to the $\operatorname{Aut}(P)$-graded algebra $\tilde{A}=A\otimes_{\mathcal{O}}\mathcal{O}\operatorname{Aut}(P)$, and we get that $\tilde{A}(P)$ is isomorphic to $\bar{N}_{A}^{\operatorname{Aut}(P)}(P)$ as $\operatorname{Aut}(P)$-graded algebras. In Section 4 we construct the extended Brauer quotient of a $\bar{G}$-graded $P$-interior algebra $A$ as mentioned above, this time with $P$ acting nontrivially on $\bar{G}$. We also discuss the exact relationship to the construction from [3]. Section 5 investigates the extended Brauer quotient of tensor products of $P$-interior algebras, in Section 6 we give an application towards correspondences for covering blocks.

Our general notations and assumptions are standard, and closely follow [8], [5] and [4].

2 The extended Brauer quotient

2.1 The construction of Puig and Zhou

We begin with a $p$-group $P$ and a $P$-interior algebra $A.$ Let $\varphi\in\operatorname{Aut}(P)$, and as in [6], we consider the $\varphi$-twisted diagonal

[TABLE]

Then the set of $\Delta_{\varphi}(P)$-fixed elements, is the following $\mathcal{O}$-submodule of $A$:

[TABLE]

Further, we consider $Q<P$ and denote by $A_{\Delta_{\varphi}(Q)}^{\Delta_{\varphi}(P)}$ the $\mathcal{O}$-module consisting of elements of the form

[TABLE]

where $c\in A^{\Delta_{\varphi}(Q)}.$ At last, we denote by $A(\Delta_{\varphi}(P))$ the quotient

[TABLE]

and we obtain the usual Brauer homomorphism

[TABLE]

If $K$ is a subgroup of $\operatorname{Aut}(P)$, it is easily checked that the external direct sum $\bigoplus_{\varphi\in K}A^{\Delta_{\varphi}(P)}$ is an algebra, while its subset $\bigoplus_{\varphi\in K}\sum_{Q<P}A^{\Delta_{\varphi}(P)}_{\Delta_{\varphi}(Q)}$ is a two-sided ideal, hence we have the following definition.

Definition 2.1** ([6]).**

The extended Brauer quotient associated to the $P$-interior algebra $A$ and the subgroup $K$ of $\operatorname{Aut}(P)$ is the external direct sum

[TABLE]

Remark 2.2**.**

Note that in this case, one deduces easily from the details given in [6, Section 3] and [7, Section 3] that $\bar{N}^{K}_{A}(P)$ is a $K$-graded algebra, and the map $\operatorname{Br}_{P}^{K}:=\bigoplus_{\varphi\in K}\operatorname{Br}_{\Delta_{\varphi}(P)}$ is a homomorphism of $K$-graded algebras. This fact will become even more transparent in the next section.

2.2 The case of $G$-interior algebras

In addition to the situation of subsection 2.1 we assume the $A$ is a $G$-interior algebra, where $G$ is a (not necessarily finite) group, and $P$ is a $p$-subgroup of $G.$ Conjugation induces the the group homomorphisms

[TABLE]

and for the subgroup $K$ in $\operatorname{Aut}(P),$ $N_{G}^{K}(P)$ denotes the inverse image of $K$ in $N_{G}(P)$. If $x\in N_{G}(P)$, we use denote by $\varphi_{x}$ the automorphism of $P$ given by $\varphi_{x}(u)=u^{x}=x^{-1}ux$ for all $u\in P.$

In this setting, we obtain some additional properties of the extended Brauer quotient (the details are left to the reader).

Proposition 2.3**.**

With the above notation, the following statements hold:

$\bar{N}^{K}_{A}(P)$* is a $K$-graded $N^{K}_{G}(P)$-interior algebra;* 2. 2)

If $K=N_{G}(P)/C_{G}(P)$, then we have the isomorphism

[TABLE]

of $N_{G}(P)/C_{G}(P)$-graded $N_{G}(P)$-interior algebras.

Proof.

1. We only need to notice that any $x\in N_{G}^{K}(P)$ verifies $u^{-1}x\varphi_{x}(u)=x.$

2. We define the $N_{G}(P)/C_{G}(P)$-graded map

[TABLE]

whose restriction to the identity component is an isomorphism. ∎

Remark 2.4**.**

Note that if $K=N_{G}(P)/C_{G}(P)$, then $\bar{N}^{K}_{\mathcal{O}G}(P)$ is just the group algebra $kN_{G}(P)$ considered with the obvious $K$-grading. Moreover, the construction of $\bar{N}^{K}_{A}(P)$ is clearly functorial in $A$, so the $N_{G}(P)$-interior algebra structure of $\bar{N}^{K}_{A}(P)$ comes from applying the construction to the algebra map $\mathcal{O}G\to A$.

3 An alternative construction

3.5**.**

The $\mathcal{O}P$-interior algebra $A$ admits an obvious $(\mathcal{O}P,\mathcal{O}P)$-bimodule structure. Consider the group algebra $\mathcal{O}[P\rtimes\operatorname{Aut}(P)],$ of the semidirect product $P\rtimes\operatorname{Aut}(P).$ This algebra is also a left $\mathcal{O}P$-module, hence it makes sense to consider the $\operatorname{Aut}(P)$-graded $(\mathcal{O}P,\mathcal{O}P)$-bimodule

[TABLE]

We may also use the isomorphism

[TABLE]

of $\mathcal{O}$-modules, which becomes an isomorphism of $(\mathcal{O}P,\mathcal{O}P)$-bimodules, by defining the bimodule structure of $A\otimes_{\mathcal{O}}\mathcal{O}\operatorname{Aut}(P)$ as follows:

[TABLE]

for $u,v\in P$ and $\phi\in\operatorname{Aut}(P)$. Then we regard $A\otimes_{\mathcal{O}}\mathcal{O}\operatorname{Aut}(P)$ as an $\operatorname{Aut}(P)$-graded $P$-algebra with $P$-action given by

[TABLE]

With the notations of Sections 2 and 3 we have:

Theorem 3.6**.**

There is an isomorphism

[TABLE]

of $\operatorname{Aut}(P)$-graded algebras, where $\tilde{A}(P)$ is the usual Brauer quotient of $\tilde{A}$.

Proof.

As the $p$-group $P$ is a normal subgroup of $P\rtimes\operatorname{Aut}(P)$, we get the decomposition

[TABLE]

If $a\otimes(1,\varphi)\in(A\otimes(1,\varphi))^{P}$, then

[TABLE]

Then $a\in A^{\Delta_{\varphi}(P)},$ and consequently

[TABLE]

is a well-defined map of $\mathcal{O}$-modules for every $\varphi\in\operatorname{Aut}(P).$ We extend this map to a $\operatorname{Aut}(P)$-graded map between these two modules and we notice that, with all the above identifications, it is actually an isomorphism of algebras. ∎

Remark 3.7**.**

We often use subgroups of $P$, and we obviously have the isomorphism

[TABLE]

of $K$-graded algebras, for any subgroups $Q\leq P$ and $K\leq\operatorname{Aut}(Q).$

4 The extended Brauer quotient of a group graded algebra

In this paragraph we set $\bar{G}:=G/H,$ where $H$ is a normal subgroup of the finite group $G,$ $P$ is a $p$-subgroup of $G$, and let

[TABLE]

for some $H$-interior $G$-algebra $B$, so $A$ is the $G$-interior $\bar{G}$-graded algebra induced from $B$.

The following lemma says that we restrict ourselves, without loss, to a certain subgroup of $\operatorname{Aut}(P)$.

Lemma 4.8**.**

Let $\varphi\in\operatorname{Aut}(P)$, and let $O(\bar{x})$ be the orbit of $\bar{x}\in\bar{G}$ under the action of $\Delta_{\varphi}(P)$ on $\bar{G}.$ If $\mid O(\bar{x})\mid\neq 1$ then $(\bigoplus_{\bar{z}\in O(\bar{x})}B\otimes z)(\Delta_{\varphi}(P))=0.$

Proof.

Consider the element $a=\displaystyle\sum b_{z_{i}}\otimes z_{i}$ such that $u^{-1}a\varphi(u)=a.$ Since the elements $z_{i}$ are all representatives of the classes of an orbit, we can choose them such that for any $u\in P$ we obtain $u^{-1}z_{i}\varphi(u)=z_{j}.$ It follows that $b_{z_{i}}^{u}=b_{z_{j}}$, and then there is one element, say $b_{z},$ such that $b_{z}^{u}=b_{z_{i}}$ for any $i$ and any $u\in P.$ Hence

[TABLE]

where $Q$ is the stabilizer of $b_{z}\otimes z$ in $P.$ ∎

4.9**.**

The above lemma gives the motivation to introduce two subgroups of $\operatorname{Aut}(P)$, because it implies that $(A\otimes\phi)(P)=0$ for $\phi\in\operatorname{Aut}(P)$ not satisfying $\overline{\phi(u)}=\overline{u^{g}}$ in $\bar{G}$, for some $g\in G$. So let

[TABLE]

and

[TABLE]

Denote also

[TABLE]

and let $\tilde{A}:=A\otimes_{\mathcal{O}P}\mathcal{O}[P\rtimes K]$ as in Section 3.

Finally, let $N^{K}_{\bar{G}}(\bar{P})$ denote the subgroup of $N_{\bar{G}}(\bar{P})$ whose elements define an element of $K$ and let $U$ be the inverse image of $N^{K}_{\bar{G}}(\bar{P})$ in $G.$ Also let $U^{\prime}$ be the inverse image of $C_{\bar{G}}(\bar{P})$ in $G$. Observe that $N^{K}_{G}(P)=N_{G}(P)$ and $N^{K_{1}}_{G}(P)=N_{G}(P)\cap U^{\prime}$.

Lemma 4.10**.**

The group $\operatorname{Aut}_{\bar{1}}(P)$ is a normal subgroup of $\operatorname{Aut}_{\bar{G}}(P)$, hence $U^{\prime}$ is normal in $U.$ Furthermore, we have the isomorphisms

[TABLE]

Proof.

If $\varphi_{1}\in\operatorname{Aut}_{\bar{1}}(P)$ then $\overline{\varphi(u)}=\bar{u}$ for all $u\in P.$ Hence, if $\varphi\in\operatorname{Aut}_{\bar{G}}(P)$ with $\overline{\varphi(u)}=\overline{u^{\bar{g}}},$ we get

[TABLE]

Further if $x\in\bar{g}$ then $\overline{\varphi(u)}=\overline{u^{g}}=\overline{u^{x}}$ and then $\overline{g^{-1}x}\in C_{\bar{G}}(\bar{P}).$ With all of the above, the map

[TABLE]

gives the first isomorphism. The second isomorphism is obvious. ∎

We will denote by $\tilde{\varphi}$ the image of $\varphi$ in the quotient group $\operatorname{Aut}_{\bar{G}}(P)/\operatorname{Aut}_{\bar{1}}(P)$.

Theorem 4.11**.**

The algebra $\bar{N}^{\operatorname{Aut}(P)}_{A}(P)$, as constructed in 2.1, is the $U/U^{\prime}$-graded $N_{G}(P)$-interior algebra $\bar{N}^{K}_{A}(P)$ with identity component the $N^{K_{1}}_{G}(P)$-interior algebra

[TABLE]

and for any $\tilde{g}\in U/U^{\prime}$ (corresponding to $\tilde{\varphi}$), the $\tilde{g}$-component is

[TABLE]

where $\varphi_{\bar{z}}\in\tilde{\varphi}$ satisfies $\overline{\varphi_{\bar{z}}(u)}=\overline{u^{z}},$ for any $u\in P.$

Proof.

By Lemma 4.8, we obtain the following decomposition of the extended Brauer quotient

[TABLE]

where in the second sum $\tilde{\varphi}$ corresponds to $\tilde{g}.$

We see that, for any $\tilde{g}$ and any $\bar{z}\in\tilde{g},$

[TABLE]

for any $x\in U^{\prime}.$ Then

[TABLE]

The fact that this algebra is $N_{G}(P)$-interior is immediate since for any $x\in N_{G}(P)$ the element $1\otimes x$ is $\Delta_{\varphi_{\bar{x}}}(P)$-invariant. ∎

Remark 4.12**.**

1. The fact that in the above theorem every $\tilde{g}$-component of $\bar{N}_{A}^{K}(P)$ is a direct sum suggests that this algebra actually has a finer grading than stated. Indeed, it is not difficult to see that $\bar{N}_{A}^{K_{1}}(P)$ is graded by the group

[TABLE]

and in general, $\bar{N}_{A}^{K}(P)$ is graded by the group

[TABLE]

2. Applying the construction to the group algebra $\mathcal{O}G$ yields $\bar{N}_{\mathcal{O}G}^{K}(P)=kN_{G}(P)$. The map

[TABLE]

where $\phi_{g}(u)=gug^{-1}$, is a group homomorphism with kernel $C_{H}(P)$. The $\bar{G}$-graded algebra map $\mathcal{O}G\to A$ induces by functoriality the $\tilde{K}$-graded algebra map

[TABLE]

3. Observe finally that the construction of $\bar{N}_{A}^{K}(P)$ does not require the $G$-interiority of $A$. We only need a $\bar{G}$-graded algebra $A$, a group homomorphism $P\to\bar{G}$ inducing a $\bar{G}$-grading on the group algebra $\mathcal{O}P$, and a homomorphism $\mathcal{O}P\to A$ of $\bar{G}$-graded algebras.

4.13**.**

Next, we establish the connection between $\bar{N}_{A}^{K}(P)\simeq\tilde{A}(P)$ and the extended Brauer quotient $\bar{N}^{K_{1}}_{B}(P)$ of the $H$-interior $G$-algebra $B$, introduced in [3]. Recall that $\bar{N}^{K_{1}}_{B}(P)$ is an $N_{H}^{K_{1}}(P)$-interior $N_{G}(P)$-algebra constructed formally as in Section 1 above. One can easily see from the definition in [3, Section 2] that $\bar{N}^{K_{1}}_{B}(P)$ is actually a $K_{1}$-graded $N_{G}^{K_{1}}(P)$-interior $N_{G}(P)$-algebra.

Let $Q:=P\cap H$. Then, as in Section 3, let

[TABLE]

Proposition 4.14**.**

The $\mathcal{O}$-module $\tilde{B}$ is a $\mathcal{O}\Delta(P\times P)$-module via

[TABLE]

for any $u\in P,$ $b\in B$ and $\varphi\in K_{1}.$ Furthermore, we have the isomorphism

[TABLE]

of $K_{1}$-graded $N^{K_{1}}_{G}(P)$-interior $N_{G}(P)$-algebras with identity component $B(P)$.

Proof.

It is clear that for any $\varphi\in K_{1}$ we have $\varphi(u)\in Q$ for any $u\in Q,$ hence $K_{1}$ acts on $Q$ and $\tilde{B}$ is a well-defined $\Delta(P\times P)$-module and we have

[TABLE]

For any $\varphi\in K_{1}$ the map

[TABLE]

is an isomorphism of $k$-vector spaces. The direct sum of these maps is the required algebra isomorphism. ∎

Remark 4.15**.**

1. According to Theorem 3.6 and Theorem 4.11, we have the decompositions

[TABLE]

The above statements show that the $N_{G}^{K_{1}}(P)$-interior algebra $\tilde{B}(P)$ can be identified with a unitary subalgebra of $\tilde{A}(P)$, and even of $\bar{N}_{A}^{K_{1}}(P)$, such that the $N_{G}(P)$-action and the $K_{1}$-grading are preserved. For the particular case of the $H$-interior $G$-invariant group algebra $B=\mathcal{O}H$, the component $\bar{N}^{K_{1}}_{B\otimes_{H}U^{\prime}}(P)$ is the $N_{G}(P)$-algebra studied in [2, Section 5].

2. The Brauer quotient $B(P)$ of $B$ is a $C_{H}(P)$-interior $N_{G}(P)$-algebra. The argument of Proposition 2.3 implies that the induced algebra

[TABLE]

is isomorphic to a $\tilde{K}$-graded subalgebra of $\bar{N}_{A}^{K}(P)$, while

[TABLE]

is isomorphic to a $\tilde{K}_{1}$-graded subalgebra of $\bar{N}_{B}^{K}(P)$.

5 Tensor products of algebras

Recall that if $A$ and $A^{\prime}$ are two $G$-graded algebras, then the diagonal subalgebra of the $G\times G$-graded algebra $A\otimes A^{\prime}$ is the $G$-graded subalgebra

[TABLE]

The following result is an extension of [6, Proposition 3.9]

Theorem 5.16**.**

Assume that $A$ and $A^{\prime}$ are two $G$-interior algebras such that $A^{\prime}$ has a $P\times P$-invariant $\mathcal{O}$-basis, and let $K$ be a subgroup of $\operatorname{Aut}(P).$

1)* There is an isomorphism*

[TABLE]

of $K$-graded $N_{G}^{K}(P)$-interior algebras.

2)* Assume in addition that, as $K$-graded $N^{K}_{G}(P)$-interior algebras,*

[TABLE]

Then

[TABLE]

as $K$-graded $N_{G}(P)$-interior algebras.

Proof.

1. We consider the $K\times K$-graded $N^{K}_{G}(P)$-interior algebra

[TABLE]

whose diagonal subalgebra

[TABLE]

is an $N^{K}_{G}(P)$-interior $K$-graded algebra. Due to the inclusion

[TABLE]

we obtain an $\mathcal{O}$-module map

[TABLE]

sending $a\otimes a^{\prime}$ to $\overline{a\otimes a^{\prime}}.$ If $c\in A^{\Delta_{\varphi}(Q)}$ and $c^{\prime}\in(A^{\prime})^{\Delta_{\varphi}(R)},$ for some subgroups $Q$ and $R$ of $P$ then

[TABLE]

This determines an $\mathcal{O}$-module homomorphism

[TABLE]

for every $\varphi\in K.$ The direct sum of all these homomorphism is a $K$-graded algebra homomorphism between $\Delta(\bar{N}_{A}^{K}(P)\otimes_{k}\bar{N}_{A^{\prime}}^{K}(P))$ and $\bar{N}^{K}_{A\otimes_{\mathcal{O}}A^{\prime}}(P),$ which is in fact an isomorphism of interior $N^{K}_{G}(P)$-algebras since by our assumptions we have

[TABLE]

2. By the additional assumption we obtain

[TABLE]

We define the $k$-linear map

[TABLE]

for every $\varphi\in K.$ The sum of these maps determine the required isomorphism of $K$-graded interior $N_{G}(P)$-algebras between $A(P)\otimes_{k}\bar{N}_{A^{\prime}}^{K}(P)$ and $\Delta(\bar{N}_{A}^{K}(P)\otimes_{k}\bar{N}_{A^{\prime}}^{K}(P)).$ ∎

Remark 5.17**.**

1. Statement 2) of the previous theorem applies in the situation of [6, Proposition 3.8]. More precisely, let

[TABLE]

for an indecomposable endopermutation $\mathcal{O}P$-module $N$, such that $A(P)\neq 0$. Let $Q\leq P$, and let $\delta$ be the unique local point of $Q$ on $A$. Let $K:=F_{A}(Q_{\delta})$. Then [6, Proposition 3.8] says that there is an isomorphism

[TABLE]

of $N^{K}_{P}(Q)$-interior $K$-graded algebras.

2. Assume in addition that $A^{\prime}$ is $\bar{G}$-graded $G$-interior as in Section 4, and has a $P\times P$-invariant $\mathcal{O}$-basis consisting of $\bar{G}$-homogeneous elements. Then, by Remark 4.12, the isomorphism in Theorem 5.16. 2) is in fact an isomorphism of $\tilde{K}$-graded $N_{G}(P)$-interior algebras.

6 A correspondence for covering points

In this section we establish a correspondence between covering points in the case of a $G$-interior algebra that has a stable basis.

6.18**.**

We keep the notations of Section 4, and we assume that the $G$-interior $\bar{G}$-graded algebra $A:=B\otimes_{\mathcal{O}H}\mathcal{O}G$ has a $G\times G$-stable $\mathcal{O}$-basis consisting of $\bar{G}$-homogeneous elements. Further, we assume that $A$ is projective regarded as a left and as a right $\mathcal{O}G$-module. By these assumptions, $B$ is an $H$-interior permutation $G$-algebra, and it is projective both as a left and a right $\mathcal{O}H$-module.

6.19**.**

We fix a normal subgroup $N$ of $G$ that contains $P$ and a point $\beta$ of $N$ on $B$ with defect group $P.$ Then our assumptions and [3, Theorem 3.1 ] imply that $\tilde{\beta}:=\operatorname{Br}_{P}(\beta)$ is a point of $N_{N}(P)$ on $\tilde{B}(P)$ with defect group $P.$

We adopt here the definition of covering points from [1]. We say that the point $\alpha$ of $G$ on $A$ covers $\beta$ if $\alpha$ has defect group $P$ and for any $i\in\alpha$ there is an idempotent $j_{1}\in A^{N}$ that lies in the conjugacy class of $\beta$ and there is a primitive idempotent $f\in A^{N}$ belonging to a point with defect group $P$ such that $j_{1}f=fj_{1}=f$ and $if=fi=f.$

Clearly in this case we consider a particular setting in which a defect group of the points that are covered coincides with a defect group of the points that cover the given ones.

Now we can state our result.

Theorem 6.20**.**

The Brauer homomorhism

[TABLE]

determines a bijective correspondence between the points of $A^{G}$ with defect group $P$ that cover $\beta$ and the points of $\tilde{A}(P)^{N_{G}(P)}$ with defect group $P$ that cover $\tilde{\beta}.$

Proof.

Theorem 3.6 and [6, Proposition 3.3] already provide a bijection between the points of $G$ on $A$ and the points of $N_{G}(P)$ on $\tilde{A}(P)$ with the same defect group $P.$ Even more, this bijection coincides with the bijection determined by the epimorphism given by the restriction of the Brauer homomorphims

[TABLE]

Since $N$ is normal in $G$, hence $N_{N}(P)$ is also normal in $N_{G}(P)$, the fact that this bijection restricts to a bijection between the points that cover $\beta$ and $\tilde{\beta}$ is an easy verification given by [1, Theorem 3.5]. ∎

References