Counting trees and

nilpotent endomorphisms

based on Tom Leinster's

arXiv 1912 12562

H MinnesotaCombinatoricsSeminarMar 27 2020

Vic Reiner

1 Cayley's formula countingtreesreformulations

2 TransientsfrecurrentsJoyal's proof

3 Fitting's lemma

4 The Herstein Theorem

counting nilpotent

5 Leinster's proof

1 Cayley's treeformulareformulations

THEOREM Borchardt 18SoCayley 1889

trees on vertex set

I 43 ngn

2

e g n 12

12 76h I 9I O 4 2

8 ly 5

1 413There are 12 of these

I l

n n2

A reformulation

THEOREMvertex rootedtrees on Cng

n n2

nn i

6 root

g22 a

7

5

3 21

So there are 12M of these

Another reformulation

THEOREMeventual constant n i

end eggsn

because there is an easy bijection

Tinted TFcp YyEat go.EIE.ee5 7 T 5

z th f 31 1

rooted tree eventuallyconstant f

Or equivalentlysince there are n

endofunctions f Ln In total

THEOREM

For anyfinite setX

prob f X X Iisoonesrefutfly X

since if h X

then LHS TE In

2 TransientsfrecurrentsJoyal'sproof

11981

One more reformulationTHEOREM

h 2 n

fvertebmtesonlnlf n.n.inn

treeswith a choice

mothiestEEE Earn Itbijectsthese all

taiga.si sl

e s tail a

g tEs Est I in5 12 6 87

31 1 3 in 5

vertebrate endofunction

Joyalvertebrates onlntfe fqffgfugg

gnsg.EE

Itoai's5

HS 3 th

head tail spinal R010 4 2 90 cord

611g if I spinal y

Ifnerves

vertebrates only

IdecompositionsCns RUTwithalinearorderonRand Ta collectionoftreesrooted at elements of R

Joyalvertebrates onlmfe ifqffofy.gg

tail

g tIs f5 f

recurrentR

HS 3 m ya jro.orgelements

T i 199head tail spinal R 12 1 6 87 f010 4 2 90 cord 7th transient

I 3 n 5 elements

6 18 2zspinaly

Ifnerves

vertebrates only endofunctionsoning

Itdecompositions decompositions

Eid RUT In RUT

withalinearorderonR withapermutationotR

andTacollectionoftreesandTacollectionottrees

rootedatelementofRrootedatelementsofR

Needforevery subset REM abjection

hwan R e Pennotutggons

So justpick a reference linear order

Crais r R for R and biject

croai.ro ro rpefro II IIEhead tail 24 9 10 oIY010 4 2 90 10 4 29

a a4 2209

head tail A010 4 2 90 4 270 9I p i 999lg l 7

121 68734h45 3 in 5

3 Fitting's lemma 4930 s

X a finite dimensional vector spaceand f X X in End X

gives riseto a unique

f stable decomposition

X V Wor O

with ftp.nvertible flwnilpotentc Aut N e NilpN

GUV

Fitting's lemma

X a finite dimensional vectorspace

and f X X in End X

gives riseto a unique

f stable decomposition

X V to W0

with Huinvertible f nilpotent

pref These chainsstabilize in s dim steps

X ZimCf ZimF z imCfd

o E KerCf EkerCI E Kerffo

because anyequalitypersiststhereafter

Fitting's lemma

X a finite dimensional vectorspace

and f X X in End X

gives riseto a unique

f stable decomposition

X V to Wo

with ftp.nvertible f nilpotent

proof Once they stabilize

X Zim Zim F z imCfd

Gog e

µdins sum todimXvia rank nullityformula in kerffo

I

4 The Fine Herstein Theorem

Recall Cayley's Theorem was

equivalent to sayingfor all finitesetsX

Bob f f X Xisionrefanhafly

THEOREM Tune Herstein 1958

For allfinitevectorspaces X GoXEFoonlinearmap

Prob f X X Xis eventuallyconstanti e nilpotent

In other words

fnmikpstatfne.FI tie gunD

5 Leinster's proof of Fine HersteinThen

To prove

pwbffsefnipdo.FI

Leinstergives a bijection

Nilp x x X Endall linearnilpotent maps f X Xlinearmaps N X

711

But analogous to Joyal's choice of areference linear order on each subsetR C Cn

he needs a choiceforeverysubspaceVc X of

a referenceorderedbasisCuv2 ikotV

a reference complementspaceVt with

X V Vtnocounterpart forRE Inwhere my R u Rtforces Rt In R

GOAL A bijection

NilpCx xX End X

a bijection 11 Fitting'shere 1 Lemma

would Isuffice Y

decompositions

E.ainkeni.ws

REVISED GOAL A bijection

Nilpex xx III IEEEI

Given N v e NilpCX x Xlet k smallestpowerwith NKG.to

and let V span v Nfl Ntv NkIv

an N stablesubspace annihilatedbyNh

since N actsnilpotently on V its minimal

polynomial is some powerof X dividingXkso equalto Xk bydefinition of k

Hence Cr Nlr NY Nk f is an ordered

basis for V and we can define geAutN byVi Va b n Vk 7 thechosen

g I I I I orredterfendebasis

v NCDNTD Nk t for

REVISED GOAL A bijection

niipcxxxt.IEo EIwonI

Given Ny we found V ge Autw

For the rest consider N acting on

X V to

v NCD NkIvrefthenfheosen

any'tv O Nuit

Considering N acting on

X V to Vv NCD N t

b

v OyI an arbitraryan arbitrary linear map

nilpotentinNilpff VII V

So far we achieved

CN v i X V Vt

Hikari X fifty h

whichlooksalmost but notquiterightsince we really want

go.infooaIYsnevipcwl

LINEAR ALGEBRA FACT

Tee 3 KEEFEP W graphEl

7197 neutalwaith an isomorphism W

itw

U III of

Vreal

that

So now we can fix

Ny X V Vt

t.EE9n

iIiinDqreiIiisiiisrarunqNuwENilpVtbythecorresponding

WENilplw

Nigf.fm FneNilplwl

andcheck it's all reversible 17

REMARKS

Leinster's preprint is

only 5 pagesC

beautifully writtenhas more historyand useful comments

His proof should lend itself

to more geometry maybeofnilpotent cone

Nilp X c EndCX

Thanks for

yourattention

f and contact me or ChrisFraser

if you would like tospeak