Representation-finite selfinjective algebras of class <span class="mathjax-formula">$D_n$</span>

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C ^OMPOSITIO M ^ATHEMATICA

C ^HRISTINE R ^IEDTMANN

Representation-finite selfinjective algebras of class D

_n

Compositio Mathematica, tome 49, n^o2 (1983), p. 231-282

<http://www.numdam.org/item?id=CM_1983__49_2_231_0>

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231

REPRESENTATION-FINITE SELFINJECTIVE ALGEBRAS OF CLASS Dn

Christine Riedtmann

1. Introduction

In this _paper,we complete ^theclassification of finite-dimensional, ^selfinjective, representation-finite algebras ^{over an}algebraically ^closed^field

k. If such an algebra îsconnected, ^{we can}âssociate^withîtâDynkin- graph ^L1⁼An, Dn, E6, E7, ôrEg, ^the^treeclass of A ([5],2). ^The^classifi-

cation has been carried out in [5] ^foralgebras ^of^tree^classAn ^{and in}[2]

for algebras ôf^tree^classE6, E7, ândEs âs^wellâs^forâ^{class of}algebras

of tree class Dn. ^Wegave ^anexplicit description ^of^theAuslander-Reiten

quivers ^foralgebras ^of^tree^classD. ⁱⁿ[6]. ^Here^we^will^determine^how

many non-isomorphic ^basicalgebras ^of^tree^classDn give ^rise^to^agiven

Auslander-Reiten quiver. Throughout ^thearticle, ^we^assume^the^{field k}

to be algebraically ^closed.

Let L1 be one of the Dynkin-graphs An, Dn, E6, E7, ôrEg, ^{and let 24} be the corresponding translation-quiver. ^Weâssociate^withâ^{subset W}

of vertices of 24 a translation-quiver (24)w ^{in the}following way. The

underlying quiver ôf(24)w îsôbtainedby adding ^to^ZL1â^vertex^{c* and}

the two arrows c - c* and c* ~ 03C4-1c for every ^cin W. We take the translation of (Z0394)G ^to^{be the}translation of ZL1 on the common vertices and to be undefined on the vertices c*. A set W is called a configuration

of (24)w îsârepresentable translation-quiver [2]; i.e., îf(Z0394)G ^satisfies

the conditions listed in [1], ^{2.8. If}^L1^ranges^over^allDynkin-graphs, W

over all configurations ôfZ0394, ândÎIôverall non-trivial admissible

automorphism groups of (Z0394)G, ^the^residuequivers (Z0394)G/03A0 provide â complete ^listôfAuslander-Reiten quivers ôffinite-dimensional, basic,

connected k-algebras ^which^arerepresentation-finite ^andselfinjective,

but not equal ^{to k}([2], 1.3). ^Twotranslation-quivers (Z0394)G/03A0 ând (Z0394’)G’/03A0’ âreisomorphic îfândonly ^{if there}îs ânisomorphism

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f:Z0394 ~ Z0394’ ^such^thatG’ = fG ^and

03A0’ = f03A0f-1.

^Inparticular, ^L1’

equals ^L1.

In case L1 ⁼A", E6, E7, ^orE8, any ^twobasic algebras ^with^agiven

Auslander-Reiten quiver (Z0394)G/03A0 ^areisomorphic. ^{Our main}^result^is

the following:

THEOREM: Let W be a configuration of ZD n’ ^and^let

03A0 ~ {1}

^be^an

admissible automorphism group of (ZDn)G.

(a) ^{I n}^case^{char k}⁼²^{and n}⁼3m for ^someinteger ^m,^andif in ^addition

le is 03C4(2m-1)Z-stable and il ⁼03C4(2m-1)Z, there ^areexactly ^twoisomorphism

classes of ^basick-algebras ^withAuslander-Reiten quiver (ZDn)G/03A0.

(b) In all other cases, any ^twobasic k-algebras ^withAuslander-Reiten quiver (ZDn)G/03A0 ^areisomorphic.

By -r(2m - l)Z we denote the infinite cyclic ^groupgenerated by ^03C42m-1.

Notice that ânalgebra ^withAuslander-Reiten quiver (Z0394)G/03A0 îs^neces- sarily connected, selfinjective, ândrepresentation-finite.

Let be a basic k-algebra ^withAuslander-Reiten quiver F,, ^{and let} ind be the full subcategory ^{of the}category ^mod ^offinitely generated

A-modules whose objects ^arespecific representatives ^{of the}isomorph-

ism classes of indecomposable ^modules.^{Then is}^calledstandard if ind A is isomorphic ^to^themesh-category k(F,) ([1], 5.1). ^Part(a) ^of^our

theorem provides âlarge family ôfnon-standard algebras. Înfact, ^we obtain one for each isomorphism ^classôf03C4(2m-1)Z-stable configurations

of ZD3m, ^orequivalently ^{for each}configuration ^ofZAm-1 ([6],6). ^For

all such non-standard algebras A, ^we^willdescribe ind A by ^itsquiver

and relations.

Let ^usexplain for which cases the theorem was proved ⁱⁿ[2]. ^An

admissible automorphism group of (ZDn)G îsgiven by ânâdmissible automorphism group of ZDn stabilizing ^{G. The}admissible automorph-

ism groups ÎIof ZDn ^were^describedⁱⁿ[4], ^{4.2: if}ÎIîsnon-trivial, ^{it is} generated by 03C4r03C8 ^for^some^{r ~}1, where 03C8 îsânautomorphism ôfZDn

with a fixed point. ^In[2], 1, ^wegave ^aproof ^forpart (b) of the theorem in case lI is generated by 03C4r03C8 ^with^{r ~}2n - 3. We now describe the

configurations W ôfZDn ^whichâdmitânautomorphism 03C4r03C8 ^with

1 ~ ^r 2n - 3. Representatives ^{of the}^twoisomorphism ^classesôf^configurations ôfZD4 âredisplayed ⁱⁿ[2], 7.6, ândthey clearly ^do^not

admit such an automorphism. Let 0 ^be^theautomorphism ôfZDn ^which exchanges (p, n - 1) ând(p, n) ^forêachp and fixes all other vertices, where we use the coordinates introduced in [5], 1.3 for the vertices of

ZDn. ^The^setôf^vertices(p, q) with q ~ n - ¹ôfâ~-stable configuration

G consists of the 03C4(2n-3)Z-orbits of (i, ^{n -}1) ^and(i, n) ^for^someinteger ⁱ

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([2], ^1.6ôr[6], 4), ând^thus2n - 3 divides r for any automorphism 03C4r03C8 stabilizing G. Let W be a ~-unstable configuration ôfZDn for n ~ 5, ând

assume 03C4r03C8 stabilizes G, ^{where 1}^~^r ^{2n -}^{3. The}^setof vertices (p, q)

in W with q ~ n - 1 ^consists^{of three}03C4(2n-3)Z-orbits ([2], 1.6 ^or[6], 4).

Therefore, 2n - 3 and hence n must be divisible by 3, say n ⁼3m, ^and either r ⁼2m - 1 or r ⁼2(2m - 1). ^Since03C42n-3 stabilizes

G, 03C83

^does^as

well, ^andthus 03C8 ^{is the}identity. ^Tosummarize, ^we^have^toprove the theorem for basic algebras ^withAuslander-Reiten quiver F, =

= (ZD3m)G/03A0, ^{where W}îsâ03C4(2m-1)Z-stable configuration ôfZD3m ând either Il ⁼03C4(2m-1)Z ôr^{II =}7:2(2m-l)Z.

Let be such an algebra, ^{and let}^{n :}(ZD3m)G ~ 0393 be the canonical map. In ^caseÎI⁼7:2(2m-l)Z, ^weprove the theorem by constructing âÎI-

invariant well-behaved functor F:k((ZD3m)G) ~ ind ; i.e., ^a^k-linear functor F with Fx ⁼03C0x for _everyvertex x of (ZD3m)G, ^such^that

Fi: 03C0x ~ ny is ^anirreducible morphism ^{in ind} ^for^the^canonical^im-

age â.in k«ZD3m)cc) ôfevery arrow (X : x ~ y in (ZD3m)G, ^{and such}^that F(g03B1) = ^Fi^foreach g ⁱⁿÎI([5], 2.5). ^Suchâ^functor^Fînducesâ^well-

behaved functor

which is an isomorphism ([5], 2.5). The construction of _{F goes}along ^the

lines of the corresponding construction in the case An ([5], 4). ^{In par-} ticular, ^we^need^someinformation about morphisms ⁱⁿk((ZD3m)B),

which we collect in chapter ^{2. In}fact, ^weprovide ^ak-basis for

k((ZDn)B)(x,y) for any ^twovertices x and _y,where W is a 0-unstable configuration ^ofZDn, ^for^{n ~}^5.

In the remaining ^caseÎI= 03C4(2m-1)Z, ^we^defineânîdeal^Jⁱⁿ^thepath- category kL1, ^where^L1⁼

(ZD3m)B/03C4(2m-1)Z,

ând^we^{show that}^{ind A}îs isomorphic êither^to^themesh-category k(0394) ôr^tok0394/J, ^forevery al-

gebra with Auslander-Reiten quiver ^{L1. In}^case^{char k ~}2, ^{we con-}

struct an isomorphism ^tok(0394), ^whichcompletes ^theproof ^ofpart (b) ^of

the theorem. As for part (a), it suffices to show that k4/J ^isisomorphic

to ind ’ for some A’ and that k4/J ândk(0394) âre^notisomorphic îf

char k ⁼2. It is possible ^tocheck the second fact directly by showing

that some huge system ^of^linearequations ^has^no^solution.However, ^we will take a different approach, describing ^{A’ and}^the^standardalgebra A

with Auslander-Reiten quiver ^L1by quivers ^and^relations(see ^also[7])

and proving that and A’ are not isomorphic. Moreover, ^we^will^show

that ind A’ has only ^even-foldcoverings. ^Moreprecisely, the map

(ZD3m)B/03C42(2m-1)Z ~

J, which îsâcovering ôftranslation-quivers ^{for all}

s, gives ^rise^to^acovering ^functor

k((ZD3m)B)/03C4s(2m-1)Z ~

^ind^’^{if and}

only îf^sîsêven.

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1 wish to thank Brandeis University ^and^theUniversity ^of^Wash- ington ^for^theirhospitality and the Schweizerischer Nationalfonds for its support.

2. Morphisms ⁱⁿk((ZDn)B)

Let W be a 0-unstable configuration ^of7LDn, ^{for n ~}^5.By ^r^we

denote the translation-quiver (7LDn)cc. Our aim is to construct a k-basis for k(r)(x, y) for any ^twoobjects ^x^andy of the mesh-category k(r).

2.1 A vertex (p, q) ôf7LDn ôr(p, q)* ôf^r^with(p, q) E CC is called low if q n - ^{2 and}high otherwise. For _{any two}vertices x and _yof 7LD", ^we

let 03B4(x, y) be the maximal number of high projective ^vertices^onany path

in r from x or 0(x) to y ^or0(y). ^Notice^that03B4(x, z) = 03B4(x, y) ⁺03B4(y, z), provided ^there^areany paths ⁱⁿ^r^from^x^toy and from _yto z, and also that 03B4((p, q), (p’, q’)) ⁼03B4((p, n - 1), (p’ ⁺min(q’, n - 1) ⁺¹^-^{n, n -}l)).

Define a high ^vertex(p, q) ôf7LD n ^to^beW-congruent ^{if the}high ^vertex (i, j) in W with minimal i ~~ p satisfies ⁱ+ j ⁼p + q ^modulo2, ând^call (p, q) W-incongruent ôtherwise.

Let

h p, h p,

^and

1

be the three paths ^from(p, n - 2) ^to(p ⁺1, n - 2) ⁱⁿ 7LDn,

where hp

^and

hp

contain the B-congruent ^andB-incongruent high

vertex with first coordinate _p,respectively,

and lp

^goesthrough (p ⁺1,

n - 3), for any integer p. We call

hp and hp

^theW-congruent ^{and W-in-} congruent ^crenelpath starting ^at(p, ^{n -}2). ^Define^apath ^wⁱⁿ^r^to^be

stable if all vertices in w lie in ZDn- ^Call^wlow if it is stable and contains

no crenel path, ândW-congruent if it is stable and contains no W-incongruent ^crenelpath. Notice that a low path may ^startôrstop ⁱⁿâhigh

vertex and âB-congruent path ⁱⁿâhigh W-incongruent ^vertex.^Wesay that âpath f îsfree (with respect ^toB) îff is low and if ^nolow vertex

(p, q) ^off ^satisfies2p + q = 2i + j + 1 and q j for any low projective

vertex (i, j)* of r. Note that 2p ⁺min(q, n - 1) îs^constantôn"vertical lines" of 7LDn. Fig. ¹^showsâ^lowpath ^{which is}^not^free.

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DEFINITION: A path ^{w :}x ~ y in r is W-forbidden ^if^w^isW-congruent

and satisfies at least one of the following conditions:

(i) ^w^containsâ^freesubpath f : x’ ^~y’, ^where^{x’ and}y’ ârehigh, ône B-congruent ândôneB-incongruent, ândô(x’, y’) ⁼^0.

(ii) ^w^containsâproper free subpath f : ^x’^~y’, ^where^{x’ -=1=}y’ ârehigh B-congruent ând03B4(x’, y’) ⁼^0.

(iii) ^wîsfree, ^xand y âreB-incongruent, ând03B4(x, y) ⁼^1.

(iv) ^w^containsâproper free subpath f: x’ - y’, ^{where x’}and y’ âre high, ôneB-congruent ândôneB-incongruent, ândô(x’, y’) ⁼^1.

(v) ^w^contains^asubpath

hp’fhp,

^where^fis free and

A subpath ^vôf^wîsâproper subpath ^{of v ~}^w.

We call w W-admissible if it is W-congruent ^and^notW-forbidden.

Clearly, any subpath ôfâB-admissible path îsagain B-admissible.

LEMMA: (a)

If w2hpwl :

x --+ y is W-admissible, ^then

w2lpwl

^is,^too.

(b)

If fhpw is

W-admissible for ^somefree path f : (p ⁺1, ^{n -}2) ^~y, then

aflpw

^isW-admissible for any arrow 03B1:03B3 ~ ^zfor ^which

aflpw

^is^cc-

congruent.

PROOF: (a) ^Let(p, q) ^{be the}high W-congruent ^vertex^ofZDn ^{with first}

coordinate _p.Inspection ^of^the^fivepossible ^cases^showsthat, ^if

w2lpwl

is B-forbidden, ^theneither the subpath ^from^x^to(p, q) ^or^the^one^from (p, q) to y of

W2hpw,

^isB-forbidden as well.

(b) ^Assume^v⁼

aflpw

^isB-forbidden.

Since.flpw

^isB-admissible, any B-forbidden subpath ^of^v^contains

afl p,

ând^hence^we^mayâssumeâll

proper subpaths ôf^v^to^beB-admissible. Again ^we^lookâtâllpossi-

bilities separately, ândît^{turns out}that, ^whenever^vîsB-forbidden,

hpw

is B-forbidden, ^too.^We^treat^{the first}case as an example; i.e., ^we^let

v ⁼

ocflpf ’:

^{x ~}^z,^wheref ’ îsfree, ^xând^zârehigh, ôneB-congruent ând

one B-incongruent, ^andb(x, z) ⁼^{0. Then}

hpf’

^isB-forbidden of type (ii)

if x is B-congruent ândôftype (i) îf^xîsB-incongruent.

2.2 DEFINITION: Two paths ^w^{and w’}^areW-neighbors ^if^w⁼w2uw1 and w’ ⁼w2v’wl, ^{where the}^set

{v, v’}

consists either of the two paths flot ^and ôy ^from(p, q) ^to(p ⁺1, q) ^for^some(p, q) ~ B with 1 q ^{n -}2 ^orof the two paths

hp+1lp

^and

lp+1hp

^for^some^integer^p^{for which}(p, n - 1) ~ B

and (p, n) ft ^B(see Fig. 2). ^Call^w^{and w’}W-homotopic ^ifthey ^are^linkedby

a sequence ^w⁼wo, w1, ..., w, ⁼w’ of successive B-neighbors.

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Fig. 2

Note that a W-neighbor ôfâW-admissible path îsB-admissible. We call a B-admissible path ^wW-marginal îf^wîsW-homotopic ^to^some^w’

containing (p, 1) ^-(p, 2) ^-(p ⁺1, 1) ^for^ap such that (p, 1) ~ B. ^Call^w^B-

essential if it is W-admissible, ^but^notW-marginal. Compare [5], ^4.2.

We _saythat the low projective ^vertex(i, j)* lies between the low paths

w and w’ from x to y îf^w^containsâ^vertex(p, q) ând^w’â^vertex(p’, q’)

with 2p + q = 2i + j + ¹= 2p’ + q’ ^andeither q j q’ ^orq’ j q (compare [5], 5.5).

LEMMA: (a) ^Two^lowpaths ^w^{and w’}^areB-homotopic if ^andonly if ^no

low projective ^vertexlies between w and w’.

(b) Â^lowpath ^wîsW-homotopic ^to^somefree path if ândonly if ^wîs free.

PROOF: For (a), ^we^refer^to[5], 5.5, ^and(b) follows from (a) ^and^the

definition of free paths.

2.3 With _anyarrow a of r, ^weassociate its sign s(a): ^we^sets(03B1) ⁼1,

unless oc is a stable arrow of the form (p, q) ^~(p, q ⁺1) with q ^{n -}2, ⁱⁿ which case we set s(03B1) = (-1)n-q. ^For^apath ^w⁼03B1r...03B11, ^welet s(w)

= s(03B1r)...s(03B11). ^Weôbtainâ^functor^from^thepath category ôf^{r onto} the mesh-category k(r) by sending ânypath ^w^to^w⁼s(w)w, ^where^w

denotes the canonical image ôf^wⁱⁿk(r). Its kernel I, îs^{the ideal}generated by ^theêlements

03B8z = 03A3s(03B1(03C303B1))03B1(03C303B1),

where z is a stable vertex of 0393, ^the^sumîs^takenôverâllarrows a : z’ ^~z,

and 03C303B1 is the arrow rz - z’. We call I, the ideal of modified ^mesh-

relations.

LEMMA: If f : (p, n - 2) - (p’, n - 2) ^isfree, ^thenf - ^w^liesⁱⁿI,, ^where

w ⁼

lp’ - 1...lp.

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PROOF: Since f îsfree, ^w^must^befree, too, ^{and hence}^wand f âre^B- homotopic by Lemma 2.2. Clearly, differences of low B-neighbors, ând

hence of low B-homotopic paths, ^lieⁱⁿI,.

2.4 PROPOSITION: For _anytwo stable vertices x and _yof ^r,^we^have

where w runs through ^a^setof representatives of ^the16-homotopy ^classesof

,W-essential paths from ^x^toy.

REMARK: This proposition yields â^{basis for}k(F)(x, y) ⁱⁿ^case^xôry ôr both are projective, ^too.Înfact, if e.g. y ⁼(p, q)* ^for^some(p, q) ^{E CC and}^l

is the arrow (p, q) ^~(p, q)*, composition with i induces a bijection

PROOF: Let W be the vector space freely generated by ^allpaths ^from^x to y in r. Let C c S c W be the subspaces spanned by ^theW-congruent

and the stable paths, respectively, ^and^letAi ^{be the}subspace spanned by

the W-congruent paths 03B1r...03B11 for which _{ai ... a}is W-admissible. If r is the common length ^of^allpaths ⁱⁿW, ^we^have

where A is spanned by the B-admissible paths. ^We^will^define^astring ^of projections

such that the kernel of each _nilies in Is(x, y). Înaddition, ^we^{will show} that the image ôfI,(x, y) ^{under n =}03C0r ... no is the subspace ^{of A}spanned by ^theB-marginal paths ^{and the}differences of B-neighbors. ^This^will imply ôurproposition.

In order to define _{no :}W ~ S, ^we^notice^thatany path ^w^{in W}^can^be

written as

where _wiis stable and ii ^and03BAi are arrows with projective ^{head and}tail,

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respectively, ^forany i. We set

where for each i the 03B1i range ^overall stable arrows whose head is the head of _xi.By ^induction^{on m,}^the^vector^{w -}03C00w lies in Is, ^{and the} kernel of _03C00is spanned by ^such^vectors.

Let ^wbe a stable path ^and^write

where Wi ^isW-congruent ^forany i. Setting

we obtain a vector in C. By definition,

s(hp)

⁼

s(h’p) = -s(lp)

⁼¹^for^any

p, ^so

that h p

⁺

h’p - lp lies

ⁱⁿ^Is,provided ^that(p, n - 2) ~ B. ^But^we

know from [6], 6 that the second coordinate of any low point ^of^a~-

unstable configuration W ^isstrictly less than n - 2. As before, ^{we con-} clude that the kernel of _nilies in I,.

Let ^usdefine _03C0i:Ai-1 ~ Ai, ^forⁱ⁼2,..., r. ^Let^w⁼oc, ... a be ^apath

in Ai-1. If w ~ Ai, ^we^set03C0iw = w. Otherwise, ^thepath v = 03B1i ... 03B11:x~z is B-forbidden, ^whereas03B1i-1...03B11 is not. Thus v contains a unique ^W-

forbidden subpath ^of^minimallength, which includes _{ai. In}each of the

possible ^caseslisted in 2.1, ^we^define^a^linearcombination 03C8v ^{of B-}

admissible paths ^from^x^{to z,}^and^we^{show that}v - 03C8v ^{lies in}I,. ^We^set

ni W ⁼a,.... ai + 1(03C8v).

(i) ^{Assume v}^contains^a^freesubpath f : x’ ^{- z,}where x’ = (p, q) ^and

z ⁼(p’, q’) ârehigh, ôneB-congruent ândôneW-incongruent, ^with 03B4(x’, z) ⁼^{0. Set}g/v ⁼0. In order to see that v lies in Is, ît^sufficesby

Lemma 2.3 to show that

03B2lp’-1 ... lp+

^la^does,^where^03B1:^{(p, q)}^~(p ⁺1,

n - 2) and f3: (p’, n - 2) ^-(p’, q’) are arrows. Assume first p’ ⁼p + 1. The condition 03B4((p, q), (p ⁺1, q’)) ⁼⁰implies ^that^neither(p, ^{n -}1) ^nor(p, n) belongs ^to^{B. Since}ône^{of the}^vertices(p, q), (p ⁺1, q’) îsW-congruent ând

one B-incongruent, ^{we see}^thatp + q p + 1 + q’ ^modulo2, ^sothat q’

= q. Clearly, ^thepath Pot: (p, q) ^~(p ⁺^{1, n -}2) ^-(p ⁺1, q) ^liesⁱⁿIs. ^In

case p’ ⁼p + t ⁺1 for some t > 0, ^we^write

The first summand lies in I, by definition, the second and third by ^in-

duction on t.

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(ii) Îf^v^containsâproper free subpath ^{from x’}^toy’, ^where^{x’ ~}y’ âre high W-congruent ândô(x’, y’) ⁼0, ^two^{cases are}possible (see Fig. 3). În

case x = x’, y’ = (p, q), ^z⁼(p + 1, n - 2), ^and^v⁼

hp f

^for^some^free^path f, ^we^setqtv

= lpf,

^{which is}W-admissible. By (i), ^thepath

h’pf

^{lies in}^I,,^so

that

does as well. In the second case, ^wehave z ⁼y’ = (p’, q’), ^{x’ =}(p, q), ^and

v ⁼

03B2fhpv1

^for^some^free path f : (p ⁺1, ^{n -}2) ^-(p’, ^{n -}2). ^We^set 4rv ⁼

03B2flpv1,

which is W-admissible by ^Lemma2.1(b). ^{As in}^{the first}case,

v - qiv ^{lies in}I,.

Fig. 3

(iii) În^{case v}îsfree, ^xând^zârehigh B-incongruent ând03B4(x, z) ⁼1,

we must have v ⁼w, and we set nrw ⁼0. In order to see that w lies in Is, it suffices to prove that u ⁼

03B2lp’-1...lp+

^{1 a :}^{(p, q)}^-^{(p’, q’)}^does,^provided

that (p, q) ând(p’, q’) ârehigh W-incongruent ^{and there}îsexactly ône high point (i, j) ~ B ^withp ⁱ p’. Încase p ⁼i ⁼p’ - 1, ^we^have(p, q)

= (i, q) ~ B ândq’ ⁼q, since the high point (i’,j’) ^{in W}with minimal i’ ~ p’ ⁼^{i +}¹satisfies p’ ⁺q’ î’+ j’ =1= ⁱ+ j ⁱ+ q ^{modulo 2.}Indeed, consecutive high points (i, j) ând(i’, j’) ôfâ0-unstable configuration

CC satisfy i + j i’ + j’ ^modulo²([6], 4). Clearly floe : (p, q) ^~(p ⁺1,

n - 2) ^-(p ⁺1, q) ^liesⁱⁿIs. ^Letp’ ⁼p + t + 1 for some t > 0, ^and^as-

sume i + 1 p’. ^Then

lies in Is, by ^induction^on^t^{and since}

03B2hp+t

does by (i). ^In^casep’ ⁼^{i +}1, we obtain

(iv) ^Assume^v^contains^aproper free subpath ^from^x’^toy’, ^{where x’}

and y’ ârehigh, ôneW-congruent ândôneW-incongruent, ând03B4(x’, y’)

(11)

= 1. In case x ⁼x’ is B-incongruent, y’ = (p, q), ^z⁼(p ⁺1, n - 2), ^and

v

= hpf

^for^some^free^{path f,}^we^set^Ç/v

= lpf,

^{and in}^case^z⁼y’ = (p’, q’)

is B-incongruent, ^{x’ =}(p, q), ^and^v⁼^w⁼

03B2fhpv1

^for^some^free^path f : (p ⁺1, n - 2) ~ (p’, n - 2), ^we^set03C8v ⁼

Pflpvl

(Fig. 3). În^both^cases, 4fv îsB-admissible by ^Lemma2.1, ândusing (iii) it is easy ^tocheck that v - ipv ^liesⁱⁿIs.

(v) ^In^{case v}⁼

hp,fhpv1,

^where^f^is^free^and^b«p,^{n -}2), (p’ ⁺1, ^{n -}2))

= 1, ^we^set03C8v ⁼

hp,flpvl

⁺

lp,fhpvl - lp, flpvl.

^The^firstôneôf^these paths is B-admissible by ^Lemma2.1(b), ^the^secondône^because

fhpv1

^is,

and the third one by ^Lemma2.1(a). Moreover, ^we^have

which belongs to Is by (iii).

It remains to be seen that 03C0Is(x, y) ^{is the}subspace ôfÂspanned by ^the B-marginal paths ând^thedifferences of B-neighbors. Clearly, ^B-marginal paths âs^wellâsdifferences of B-neighbors ^{lie in}Is, ^since

does, ^whenever(p, n - 1) ~ ^B^and(p, n) e ^B.

As I,(x, y) ^isspanned by ^the^vectors

where _W1and _w2are paths ^from^x^to^03C4z^{and from}^z^toy for some stable

z, respectively, and where the sum is taken over all arrows a with head _z, it suffices to write _npas a linear combination of B-marginal paths ^and

differences of B-neighbors. We may ^assumethat 03C4z does not lie in B, since otherwise _nop⁼0, ^andthat y ^lies^{in S.}Similarly, ^wehave 03C0103BC ⁼⁰

if the second coordinate of z is n - 2. The proof ⁱⁿ^case^z^ishigh ^is straightforward, ^{the main}problems being ^thelarge ^{number of}possible

cases and the bookkeeping. ^Inmost cases, 03C003BC turns out to be zero. As an

example, ^we^treat^one^ofthe harder _cases,and we skip ^the^rest.

Assume z = (p + 1, q) ~ y is high B-incongruent ^and^rz⁼(p, q) ~ ^x^is B-congruent. Then y has the form

and we may ^assumethat

(12)

Let i be the length ^ofv1. Then 03C0i+1 ... 03C0103BC is either zero or a linear combination of vectors of the form

Let us assume that _V3⁼

fhp,v4,

^where^{f : (p’}⁺^{1, n -}²⁾^~(p, n - 2) ^is^free

and c5«P’, n - 2), (p ⁺^{1, n -}2)) ⁼1; i.e., ^wesuppose

hpv3

^tobe B-forbidden of type v). ^We^obtain

By ^ourassumptions, ^neither(p, n - 1) ^nor(p, n) ^{lies in}W, ^so^that 03B4((p, n - 1), (p + 1, n - 1)) = 0 ^and03B4((p’, n - 1), (p + 1, n - 1)) = 1. ^Hence

v2hp+1hpflp’v4

^is^theonly path occurring ⁱⁿv 1 which does not lie in

Ai+3. ^We^obtain

Suppose p ⁼

v2(lp+1hp-hp+1lp)v5

belongs ^toAj, ^but^{not to}Aj+1 ^for some j ^with^{i +}⁴~ j ^r,^{and let}V2 ⁼v7v6, where the length ^ofv6 is

j - ⁱ^-3. In case v6 itself is W-forbidden, ^weclearly ^have

Otherwise,

are B-forbidden of the same type, ^sinceb«P, n - 1), (p ⁺1, n - 1)) ⁼^0.

Unless they ^areB-forbidden of type (v), ^we^have03C0j+103C1 ⁼0, ^since_03C0j+1 either annihilates both summands separately, ^or

In the remaining case, there is a free path f : (p ⁺2, n - 2) ~ (p’, n - 2),

where l5«P ⁺1, n - 2), (p’ ⁺1, n - 2)) ⁼1, ^{such that}v6

= hp. f.

^Then

(13)

so that by înduction^we^mayâssumep lies ⁱⁿA, ând^henceîtîs^the difference of two B-neighbors.

Finally, ^if^rz⁼(p, q) ^does^not^{lie in W}and q ~ n - 3, nJ1 is a linear combination of vectors of the form

each of which is either the différence of two B-neighbors ^orB-marginal.

2.5 In the remainder of this chapter, ^we^derive^theauxiliary ^results

needed in the proof ^{of the}^theorem.^From^now^on,^{we assume}^that

contains the vertex (0, n - 1). ^This^condition^canalways be fulfilled by replacing by ^anisomorphic configuration. ^We^recall^thefollowing description ^{of W}^from[6], 6. ^The^set^ofhigh vertices of W consists of the t(2n - 3)Z-orbits of

for some natural numbers (including zero) nl, n2, and _n3with _{n, + n2} + n3 ^{= n -}3. There are configurations D1, D2, ^andD3 ^ofZAn1, ZAn2,

and ZAn3’ respectively, such that the set of low vertices of W is the dis-

joint ^union^{of the}^sets

For any natural number m ~ n - 2, ^theinjection

from the vertex set of ZAm ^to^thevertex set of ZDn ^is^definedby

and by requiring ^thatt/lm ^03C4m⁼^03C42n-303C8m, where 03C4 denotes the translation of ZAm ^on^the^left-hand^{side and}ZDn ^on^theright-hand side. Notice

that, for any m ^{n -}2, 03C8m ^factorsthrough 03C8m+1. ^Infact, ^we^havet/lm

= 03C8m+1 03C9m, where the injection

is given by

Representation-finite selfinjective algebras of class <span class="mathjax-formula">$D_n$</span>

C OMPOSITIO M ATHEMATICA

C HRISTINE R IEDTMANN

Representation-finite selfinjective algebras of class D

03A0’ = f03A0f-1.

03A0 ~ {1}

G, 03C83

(ZD3m)B/03C4(2m-1)Z,

(ZD3m)B/03C42(2m-1)Z ~

k((ZD3m)B)/03C4s(2m-1)Z ~

h p, h p,

1

where hp

hp

and lp

hp and hp

hp’fhp,

If w2hpwl :

w2lpwl

If fhpw is

aflpw

aflpw

w2lpwl

W2hpw,

aflpw

Since.flpw

afl p,

hpw

ocflpf ’:

hpf’

{v, v’}

hp+1lp

lp+1hp

03B8z = 03A3s(03B1(03C303B1))03B1(03C303B1),

lp’ - 1...lp.

s(hp)

s(h’p) = -s(lp)

that h p

h’p - lp lies

03B2lp’-1 ... lp+

hp f

= lpf,

h’pf

03B2fhpv1

03B2flpv1,

03B2lp’-1...lp+

03B2hp+t

= hpf

= lpf,

03B2fhpv1

Pflpvl

hp,fhpv1,

hp,flpvl

lp,fhpvl - lp, flpvl.

fhpv1

fhp,v4,

hpv3

v2hp+1hpflp’v4

v2(lp+1hp-hp+1lp)v5

= hp. f.

C ^OMPOSITIO M ^ATHEMATICA

C ^HRISTINE R ^IEDTMANN