98

数理解析研究所講究録
第2061巻 2018年 98-105

CENTRAL ELEMENTS OF THE JENNINGS BASIS AND
CERTAIN MORITA INVARIANTS

TARO SAKURAI(千葉大学大学院理学研究科

櫻井太朗)

ABSTRACT. From Morita theoretic viewpoint, computing Morita invari‐ants is important. We proved that the intersection of the center and thenth socle ZS^{n}(A) :=Z(A)\cap \mathrm{S}\mathrm{o}\mathrm{c}^{n}(A) of a finite dimensional algebra A
is Morita invariant; This is a generalization of important Morita invari‐ants, the center Z(A) and the Reynolds ideal ZS^{1}(A) .
As an example, we also studied ZS^{n}(FP) for the group algebra FP
of a finite p‐group P over a field F of positive characteristic p . Suchan algebra has a basis along the radical filtration, known as the Jen‐nings basis. We show sufficient conditions under which an element of
the Jennings basis is central and a lower bound for the dimension of
ZS^{n}(FP) for every positive integer n . Equalities hold for 1 \leq n \leq p
if P is powerful. As a corollary we have So \mathrm{c}^{} (FP) \subseteq Z(FP) if P ispowerful.

This is a report of a talk based on [Sakurai, \mathrm{a}\mathrm{r}\mathrm{X}\mathrm{i}\mathrm{v}:1701.03799\mathrm{v}2 ].

CONTENTS
1.

Introduction

1

2.

Morita invariants

2

3.

Jennings theory

3

4.

Main theorems

4

5.

Examples6

References8

1. INTRODUCTION

From Morita theoretic viewpoint, computing Morita invariants is important
to distinguish algebras that are not Morita equivalent. We show that the
DEPARTMENT OF MATHEMATICS AND INFORMATICS, GRADUATE SCHOOL OF SCIENCE,CHIBA UNIVERSITY, 1‐33, YAYOI‐CHO, INAGE‐KU, CHIBA‐SHi, CHIBA, 263‐8522 JAPANE‐mail address: tsakurai@math.s.chiba‐u.ac.jp.
2010 Mathematics Subject Classification. 16\mathrm{G}30 (primary), 16\mathrm{U}70, 16\mathrm{D}90, 16\mathrm{D}25,
20\mathrm{C}20 (secondary).

Key words and phrases. Morita invariant, center, socle, Reynolds ideal, p‐‐group, Jen‐nings basis, dimension subgroup.
1

99
T. Sakuraiintersection of the center and the nth right socle

ZS^{n}(A) :=Z(A)\cap \mathrm{S}\mathrm{o}\mathrm{c}^{n}(A)is Morita invariant for a finite dimensional algebra A (Theorem 2.2) in Sec‐
(1.1)

tion 2. This is a generalization of important Morita invariants, the center
Z(A) and the Reynolds ideal ZS^{1}(A) . The other way of generalization is
known as the Külshammer ideals or the generalized Reynolds ideals for asymmetric algebra; For more details we refer the reader to the survey byZimmermann [17].
For the group algebra FG of a finite group G over an algebraically closed
field F of positive characteristic p , the dimension of the center Z(FG) and
the Reynolds ideal ZS^{1}(FG) , respectively, equal the number of irreducible
ordinary characters k(G) and the irreducible modular characters \ell(G) (see[9, Lemmas 3 and 4(See also (4.3) for n=2. ) Moreover the conjugacy
class sums form a basis for the center and the p‐‐regular section sums forma basis for the Reynolds ideal (see [5, Satz \mathrm{D} ]; see also [9, Lemma 3] and[11, Theorem 1
These are summarized in Table 1.1 where t denotes the

Loewy length of FG.

TABLE 1.1. What is known about ZS^{n}(FG) .

ZS^{t}(FG)ZS^{n}(FG)ZS^{2}(FG)ZS^{1}(FG)dimension (representation‐theoretic)

basis (group‐theoretic)k(G)conjugacy class sums

unknown

unknown\displaystyle \ell(G)+\sum_{S:} simple dimExt1 (S, S)

unknown\ell(G)*p‐regular section sums

As ZS^{n}(FG) is a generalization of such, we want to know what is thedimension and what a basis can be. One of manageable examples to compute

socle series is the group algebra FP of a finite p‐‐group P over a field F of
positive characteristic p . Jennings constructed a basis of FP along theradical filtration (Theorem 3.5) and it follows that radical series coincides
with socle series (Theorem 3.6). Such basis is known as the Jennings basis(Definition 3.7). To give a lower bound for the dimension of ZS^{n}(FP) we ask
a question: When is an element of the Jennings basis central? In Section 4
we give sufficient conditions under which an element of the Jennings basis iscentral and a lower bound for the dimension of ZS^{n}(FP) for every positive
integer n (Theorem 4.1). This lower bound is sharp; We also proved that theequalities hold for 1\leq n\leq p if P is powerful (Theorem 4.4). As a corollary
we have So \mathrm{c}^{} (FP) \subseteq Z(FP) if P is powerful (Corollary 4.5). Examples in
Section 5 show that the corollary is best possible.
2. MORITA INVARIANTS

Inspired by the proof of Morita invariance of the Külshammer ideals by

Héthelyi et al. [2, Proposition 5.1] (see also [16, Theorem 1] for their derived*

Except finite p‐‐groups; see Remark 4.7.
2

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Central elements of the Jennings basis

invariance), we showed that the intersection of the center and the nth right

socle ZS^{n}(A)=Z(A)\cap \mathrm{S}\mathrm{o}\mathrm{c}^{n}(A) , an ideal of the center, is a Morita invariant
for a finite dimensional algebra A over a field. Note that this is left‐rightsymmetric unlike socle. In the following Ra \mathrm{d}^{} (A) denotes the nth Jacobsonradical of A.

Lemma 2.1. Let A be a finite dimensional algebra over a field and supposee \in A is a full idempotent.
Then Ra\mathrm{d}^{} (eAe) =\mathrm{R}\mathrm{a}\mathrm{d}^{n}(A)\cap eAe for every
positive integer n.

Theorem 2.2 (Sakurai [12]). Let A and B be Morita equivalent finite
dimensional algebras over a field.

Then there is an algebra isomorphism

Z(A) \rightarrow Z(B) such that the following diagram commutes for all positive
integer n . In particular, ZS^{n}(A) are Morita invariants.

Z(A) \rightarrow Z(B)\uparow\uparow

ZS^{n}(A) \rightarrow ZS^{n}(\mathrm{B})3. JENNINGS THE0RY

As we established Morita invariance of ZS^{n}(A) in Theorem 2.2, we want
to determine the invariants for special cases. We hereafter study a groupalgebra FP of a finite p‐‐group P over a field F of positive characteristic p.
In this section we collect results of the Jennings theory.

Definition 3.1. For a positive integer i we define the ith dimension subgroup(or Jennings subgroup) of P by

D_{i} :=\{u\in P|u-1\in \mathrm{R}\mathrm{a}\mathrm{d}^{i}(FP)\}.
Remark 3.2. Although the dimension subgroups are defined ring‐theoretically,

those can be computed group‐theoretically by Theorem 3.6(iv).
Lemma 3.3.

(i) Every dimension subgroup is a characteristic subgroup.
(ii) Every successive quotient of the dimension subgroups is an elemen‐tary abelian p ‐group.

Notation 3.4. Let D_{i} be the dimension subgroups of P . For a successive
quotient of the dimension subgroups D_{i}/D_{i+1} of p‐‐rank r_{i} , we fix elements
u_{i1} , . . . , u_{ir_{\mathrm{t} } \in D_{i} such that

D_{i}/D_{i+1}=\langle u_{i1}D_{i+1}\rangle \mathrm{x}

.

.

.

\times\{u_{ir_{ $\iota$}}D_{i+1}\rangle.

Set \ell :=\displaystyle \min\{i \geq 1 | D_{i} = 1\}, $\Lambda$ := \{(i,j) | 1 \leq i <\ell, 1 \leq j \leq r_{i}\} , andM:=\{0, 1, . . . , p-1\}^{ $\Lambda$} . For $\mu$=($\mu$_{ij})\in M define(3.1)

w( $\mu$):=\displaystyle \sum_{(i,j)\in $\Lambda$}i$\mu$_{ij} and z^{$\mu$}:=\displayst le\prod_{(i,j)\in$\Lambda$}'z_{ij}^{$\mu$_{$\iota$g},
3

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T. Sakurai

where z_{ij} := u_{ij}-1 and the productFor an integer k define\displaystyle\prod' is taken in lexicographic order.

(3.2)$\mu$_{ij}=p-1 for all (i, j)\in $\Lambda$ }M_{k} := { $\mu$\in M|i\geq k\Rightarrow

and $\mu$_{k}\in M_{k} by($\mu$_{k})_{ij}=\left\{ begin{ar ay}{l} p-1&(i\geqk)\ 0&(i<k). \end{ar ay}\right.

(3.3)

Theorem 3.5 (Jennings [3]). For every non‐negative integer n we have
Ra \mathrm{d}^{} (FP)=\oplus Fz^{ $\mu$}.

$\mu$\in Mw( $\mu$)\geq n

Theorem 3.6 (Jennings [3]).
(i) The Loewy length \ell\ell(FP) of FP equals 1+w($\mu$_{1}) .
\mathrm{R}\mathrm{a}\mathrm{d}^{\ell\ell(FP)-n}(FP) for every 0 \leq(ii) FP is rigid: So \mathrm{c}^{} (FP)\ell l(FP)(iii) So \mathrm{c}^{} (FP)=\oplus Fz^{ $\mu$} for every n\geq 0.
=

n

\leq

.

$\mu$\in M

w($\mu$_{1})-w( $\mu$)<n(iv) D_{1}=P and D_{i}=(D_{\lceil i/p\rceil})^{p}[D_{i-1}, P] for every i>1.
Definition 3.7. The basis \{z^{ $\mu$} | $\mu$\in M\} of FP is said to be the Jennings
basis.
4. MAIN THEOREMSAs promised, we give sufficient conditions under which an element of theJennings basis is central and a lower bound for the dimension of the Morita
invariant ZS^{n}(FP)Z(FP)\cap \mathrm{S}\mathrm{o}\mathrm{c}^{n}(FP) for every positive integer n in=

Theorem 4.1.

Theorem 4.1 (Sakurai [12]). Let F be a field of positive characteristic p and
P a finite p ‐group. Suppose k is a positive integer that satisfies D_{k}\geq [P, P]
where D_{k} denotes the kth dimension subgroup of P (recall Definition 3.1).
Then, with Notation 3.4 and (1.1), we have
(4.1)

ZS^{n}k(FP)\displaystyle \supseteq\bigoplus_{ $\mu$\in M_{k} Fz^{ $\mu$}where n_{k} := 1+w($\mu$_{1})-w($\mu$_{k}) . In particular, for every positive integer n
we have(4.2)

ZS^{n}(FP)\supseteq\oplus Fz^{ $\mu$}.
$\mu$\in M_{k}w($\mu$_{1})-w( $\mu$)<n

Remark 4.2. Note that such k always exists: D_{2}\geq [P, P] . Note also thatthe dimension of the right hand side of (4.1) equals |M_{k}|=|P/D_{k}|.
4102Central elements of the Jennings basis
See Section 5 for concrete examples. We can show that those coincide
under the following conditions.

Definition 4.3. A finite p‐group P is said to be powerful if [P, P] \leq P^{p}
and p>2 , or [P, P] \leq P^{4} and p=2.

Theorem 4.4 (Sakurai [12]). If P is powerful then for every 1\leq n\leq p we
haveZS^{n}(FP)=\oplus Fz^{ $\mu$}.
$\mu$\in M_{2}w($\mu$_{1})-w( $\mu$)<n

Corollary 4.5 (Sakurai [12]). If P is powerful then we haveSo \mathrm{c}^{} (FP)\subseteq Z(FP) .

We give some remarks concerning these theorems in the rest of this sec‐tion.

Remark 4.6. It is known that \mathrm{S}\mathrm{o}\mathrm{c}^{2}(A) \subseteq
split‐local symmetric algebra A . This canZ(A) for a finite dimensionalbe traced back to Müller [7,Proof of Lemma 2]. (For a simple proof see, for example, [1, Lemma 2.2].)
Remark 4.7. Note that ZS^{n}(FP) can be written down explicitly for n=
1 , 2:ZS^{1}(FP)=Fz^{$\mu$_{1}}ZS^{2}(FP)=Fz^{$\mu$_{1} \displaystyle \oplus\bigoplus_{1\leq j\leq r1}Fz^{$\mu$_{1}-$\delta$_{J}
where $\delta$_{j}\in M is an element with 1 at ( 1, j) and 0 otherwise. This is due to
the Jennings theory and Remark 4.6.

Remark 4.8. Huppert raised a question that are the dimensions of Loewy

layers of FP unimodal? Negative answer is given by Manz‐Staszewski [6]and Stammbach‐Stricker [15], independently. Nevertheless positive answer

is known under certain condition; Shalev proved that the dimensions of
Loewy layers are unimodal for powerful p‐‐group if p>2 (see [13, Proposi‐tion 4.1]). Hence it is reasonable to assume powerful as the Loewy series iswell‐behaved.

Remark 4.9. Let A be a block of a finite group algebra (or a finite di‐mensional symmetric algebra) over an algebraically closed field. Okuyama

obtained the dimension of ZS^{2}(A) that

(4.3)

\displaystyle \dim ZS^{2}(A)=\dim ZS^{1}(A)+\sum_{S}\dim \mathrm{E}\mathrm{x}\mathrm{t}_{A}^{1}(S, S)

where the sum is taken over a set of representatives of isoclasses of simple
A ‐modules[8].

(See also [4, Theorem 2.1] which is written in English.)5

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T. Sakurai

Recently Otokita obtained an upper bound for the dimension of ZS^{n}(A)
that

(4.4)

\displaystyle \dim ZS^{n}(A)\leq\sum_{S}c(P_{S}/\mathrm{R}\mathrm{a}\mathrm{d}^{n}(P_{S}), S)

where the sum is taken over a set of representatives of isoclasses of simple
A ‐modules and c(P_{S}/\mathrm{R}\mathrm{a}\mathrm{d}^{n}(P_{S}), S) denote the composition multiplicity ofa simple module S in the factor module P_{S}/\mathrm{R}\mathrm{a}\mathrm{d}^{n}(P_{S}) of the projective

cover P_{S} of S [ 10 , Theorem 1.1].

5. EXAMPLESIn this section we illustrate our results by examples. In particular, theexamples show that Corollary 4.5 is best possible. In the following we con‐sider group algebras of extra‐special p‐‐groups of order p^{3} for odd prime pover a field F of characteristic p.

5.1. Extra‐special‐group p_{+}^{1+2} . Let P be an extra‐special p‐‐group oforder p^{3} and exponent p defined by

P:=p_{+}^{1+2}=M(p)=\langle a, b, c|a^{p}=b^{p}=d^{\mathrm{J}}=[a, c]=[b, c]=1, [b, a]=c\rangle
for odd prime p and set x :=a-1, y:=b-1 , and z:=c-1 . Then we canshow that

(5.1)

(5.2)

Ra \mathrm{d}^{} (FP)=\oplus Fx^{i}y^{j}z^{k}i+j\dotplus 2k\geq n0\leq ij,k<p

Z(FP)=\displaystyle \bigoplus_{0\leq k<p-1}Fz^{k}\oplus\bigoplus_{0\leq i,j<p}Fx^{i}y^{j}z^{p-1}

In particular, for p=3 , we have

(5.3)

FP\sim[x^{2}y x^{2}y zx^{2}y x^{2}z x^{2}z xy^{2}z xy^{2} yz^{2}x z^{2}y _Zxzy^{2} y^{2}z xy^{2}z y^{2}z y^{2}z y1]

which mean that ith row consists of the elements of the Jennings basis lying
in \mathrm{R}\mathrm{a}\mathrm{d}^{i-1}(FP)\backslash \mathrm{R}\mathrm{a}\mathrm{d}^{i}(FP) and bold letters show that the elements arecentral. Note that P is not powerful and x^{2}y^{2}z \in \mathrm{S}\mathrm{o}\mathrm{c}^{3}(FP) \backslash Z(FP) .
Hence the assertion of Corollary 4.5 does not hold without the assumptionthat P is powerful.
6

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Central elements of the Jennings basis
TABLE 5.1. Relevant numbers of the main theorems for P=3_{+}^{1+2}n 1 2 3 4 5 6 7 8 9\overline{\dim \mathrm{S}\mathrm{o}\mathrm{c}^{n}(FP)137111620242627}\dim ZS^{n}(\mathrm{F}P) 1 3 6 8 9 9 10 10 11\#\{ $\mu$\in M_{2} |w($\mu$_{1})-w( $\mu$)<n\} 1 3 6 8 9 9 9 9 95.2. Extra‐special‐group p_{-}^{1+2} . Let P be an extra‐special p‐‐group oforder p^{3} and exponent p^{2} defined by

P:=p_{-}^{\mathrm{I}+2}=M_{3}(p)=\{a, b|a^{p}=b^{p^{2}}=1, b^{a}=b^{1+p}\rangle
for odd prime p and set x :=a-\mathrm{I}, y:=b-1 , andz: =c-1 where c=\mathcal{U}^{1}.
Then we can show that

(5.4)

(5.5)

Ra\mathrm{d}^{} (FP)=\oplus Fx^{i}y^{j}z^{k}i+j0\leq ij,k<p\dotplus_{pk\geq n}

Z(FP)=\displaystyle \bigoplus_{0\leq k<p-1}Fz^{k}\oplus\bigoplus_{0\leq i,j<p}Fx^{i}y^{j}z^{p-1}

In particular, for p=3 , we have

(5.6)

FP\sim[x^{2}z yz^{2}x y^{2_Z}xz y^{2}xz y^{2}z x ^{2}y z ^{2}y z ^{2}y1z]where the convention is the same as (5.3). Note that P is powerful andx^{2}y^{2}z \in \mathrm{S}\mathrm{o}\mathrm{c}^{3+1}(FP) \backslash Z(FP) . Hence even if P is powerful a stronger
assertion of Corollary 4.5 that \mathrm{S}\mathrm{o}\mathrm{c}^{p+1}(FP)\subseteq Z(FP) is false in general.
TABLE 5.2. Relevant numbers of the main theorems for P=3_{-}^{1+2}n 1 2 3 4 5 6 7 8 9 10 11

dimSoc”(FP) 1 3 6 9 12 15 18 21 24 26 27\dim ZS^{n}(\mathrm{F}\mathrm{P}) 1 3 6 8 9 9 9 10 10 10 11
9
9
9
9
9
9
9
\#\{ $\mu$\in M_{3}|w($\mu$_{1})-w( $\mu$)<n\} 1 3 6 87

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T. Sakurai

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