Table Of ContentGraphs of relations and Hilbert series
8
0 Peter Cameron & Natalia Iyudu
0
2
n
Queen Mary, University of London
a
J e-mail: [email protected]
9 Abstact We are discussing certain combinatorial and counting problems related
1
to quadratic algebras. First we give examples which confirm the Anick conjecture
on theminimal Hilbert series for algebras given by n generators and n(n−1) relations
] 2
A for n 6 7. Then we investigate combinatorial structure of colored graph associated
R to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to
algebras with maximal Hilbert series are given in certain cases. As a consequence it
.
h turns out, for example, that RIT algebra may have a maximal Hilbert series only if
t components of thegraph associated to each color are pairwise 2-isomorphic.
a
m
Keywords: Quadraticalgebras; Hilbert series; Gr¨obner basis; Colored graph
[
1
v
3
1
Contents
0
3
. 1 Introduction 1
1
0
2 The Anick conjecture for n67 3
8
0 2.1 Series in general position. . . . . . . . . . . . . . . . . . . . . . . 3
: 2.2 The Anick conjecture holds for n67 . . . . . . . . . . . . . . . . 4
v
i
X 3 RIT algebras and maps of the finite set 5
r 3.1 The class of RIT algebras . . . . . . . . . . . . . . . . . . . . . . 5
a
3.2 Condition on maps and Gr¨obner basis . . . . . . . . . . . . . . . 6
3.3 Representations of the semigroup hx |x =x x i . . . . . . . . . . 7
i i i j
3.4 CombinatorialdescriptionofmapscorrespondingtomaximalHilbert
series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Toward classification of Hilbert series 11
5 Remark on the Generalized Yang–Baxter Equation for RIT al-
gebras 12
6 Acknowledgments 13
1 Introduction
Let A(n,r) be the class of all graded quadratic algebras on n generators and r
relations:
A=khx ,...,x i/id{p :i=1,...,r},
1 n i
1
n
where p = αk,jx x , αk,j ∈k.
i i k j i
k,Pj=1
We deal with an arbitrary field k of char 0. Only on the way (section 2.1)
we restrict ourself with C for a while (to get more general statement), but it
will not influence further results.
∞
These algebras are endowed with a natural filtration A = U , where
m
mS=0
U is the linear span of monomials on a of degree not exceeding m, a are
m i i
the images of the variables x under the canonical map from khx ,...,x i to
i 1 n
A and the degree of a ...a equals d. Since the generating polynomials are
i1 id ∞
homogeneous,the algebraA∈Aalsopossessesa canonicalgradingA= A ,
i
iL=0
where A is the linear span of monomials of degree exactly i. This grading has
i
a finiteness property: dim A <∞ forany i, since algebrais finitely generated.
k i
This allow to associateto the series of dimensions variousgenerating functions.
The one which reflects most straightforward properties of the algebra will be
considered here.
∞
Definition 1.1 The Hilbert series of a graded algebra A = A is the gen-
i
iL=0
erating function of the series of dimensions of graded components d =dim A
i k i
of the following shape: H (t)= ∞ d ti.
A i=0 i
P
We are going to confirm Anick’s conjecture [1] saying that a lower bound
for the Hilbert series of an algebra with n(n−1) quadratic relations given by
2
theseries 1−nt+ n(n−1)t2 −1 isattained,forthesmallnumberofvariables
(cid:12) 2 (cid:12)
(cid:12)(cid:16) (cid:17) (cid:12)
n 6 7. H(cid:12)ere the sign of modulu(cid:12)s stands for the series where nth coefficient
(cid:12) (cid:12)
equals to the nth coefficient of initial series if this is positive together with all
previous coefficients and is zero otherwise.
After notices on minimal and generic series for quadratic algebras we turn
to the main subject of our investigation. We consider subclass R(m,n) ⊂
A(g,g(g−1)), where g = n + m, called RIT algebras (it was introduced and
2
studiedinpapers[4,3,2]). ClassR(m,n)definedasconsistingofalgebraswith
presentation of the form
R=khx ,...,x ,y ,...,y i/F,
1 m 1 n
where
[x ,x ]=0,
i j
F =id [yi,yj]=0, (1)
[x ,y ]=y y ,
i j f(i,j) j
and f is a map f :M ×N −→N, M ={1,...,m}, N ={1,...,n}.
To any such algebra we associate m-colored graph with n vertices in such
a way that subgraph Γ of color i reflects the map σ : N → N defined by
i i
σ (n)=f(i,n).
i
We formulate conditions (Theorem 3.1) on the above maps which mean
that defining relations of algebra form a Gr¨obner basis, or equivalently that
the algebra has a lexicographically maximal Hilbert series. Then we attack
more subtle question on how to describe precisely the combinatorial structure
of those maps. This is done explicitly in the Theorem 3.8 for a pair of maps.
As a consequence interesting necessary condition is obtained for an algebra to
have a maximal Hilbert series. Corollary 3.9 says that Hilbert series of algebra
could be maximal only if graphs of all maps σ ,i = 1,¯m are 2-isomorphic. We
i
2
callgraphs2-isomorphic iftheybecomeisomorphicaftergluingpairsofvertices
in common cycles of length two.
Described combinatorial conditions (Theorem 3.8) also imply that algebra
is Koszul and obeys a commutator generalization of Yang-Baxter equation.
Another consequences for RIT algebras from being presented by a quadratic
Gr¨obner basis are that in this case they are Auslander regular and Cohen-
Macaulay. Hence we get combinatorial conditions on graphs sufficient also for
obeying these properties.
Inthesection4wepresentthecompletelistofHilbertseriesandcorrespond-
ing non isomorphic colored graphs for RIT algebras of rank up to 4.
2 The Anick conjecture for n 6 7
2.1 Series in general position
We remind here the proof of minimality of general series because it is essential
forthenextsection. Versionwepresentdealswiththenotionofgeneralposition
in the Lebesgue sense, so we suppose for this section that the field is k = C.
Essentially the knowledge on this matter is due to Anick [1] and explanations
insimplified formone canfindinthe surveyofUfnarovskii[17]. In [15]onecan
also find a remark on the minimality of nth component of the series in general
position in the Zarisski sense. But as it is pointed out in [17] (remark before
the theorem 3 in I.4.2), since the infinite union of proper affine varieties may
not be contained in a proper affine variety, one can not state that the minimal
series (in case it is infinite) is in general position in the Zarisskii sense. There
could be several ways to avoid this problem, for example in spirit of theorem
3, I.4.5. in [17]. But over C most natural and easy way is to use the topology
defined by the Lebesgue measure, so we present this version here.
Let we define now more precisely what is meant by an algebra in general
position in the Zarisskii and in the Lebesgue sense. Algebras from A(n,r) are
naturally labeled by the points of krn2, corresponding to the coefficients αk,j
i
of the relations. Given a property P of quadratic algebras, we say that P is
satisfied for A ∈ A(n,r) in general position in the Zarisskii sense if the set
of the coefficient vectors corresponding to those A ∈ A(n,r), which obey the
property P, is a non-empty Zarisskii-open subset of krn2. We also say that P
is satisfied for A∈A(n,r) in general position in the Lebesgue senseifthe setof
the coefficient vectors corresponding to those A ∈ A(n,r), which do not obey
the property P has rn2-dimensional Lebesgue measure zero. Since the set of
zeros of any non-zero polynomial has the Lebesgue measure zero, we see that
as far as arbitrary property P satisfied for A ∈ A(n,r) in general position in
the Zarisskii sense it is also satisfied for A ∈ A(n,r) in general position in the
Lebesgue sense. Define the minimal series in the class A(n,r) componentwise:
∞
Hn,r(t)= b ti,
min i
Xi=0
where b = min dimA , A being the ith homogeneous component in the
i i i
A∈A(n,r)
grading of A. It is not clear a priori, whether there exists an algebra A ∈
A(n,r) whose Hilbert series coincides with Hn,r. This follows however from
min
the statement below.
Proposition 2.1 For A ∈ A(n,r) in general position in the Lebesgue sense,
the equality H =Hn,r is satisfied.
A min
3
Proof. Denote the ideal generated by {p : 1 6 i 6 r} by I and its dth
i
homogeneous component by I . Obviously
d
I =span {up v :u,v ∈hx ,...,x i, degu+degv =d−2}.
d k i 1 n
Here hx ,...,x i stands for the free semigroup generated by {x ,...,x }. Let
1 n 1 n
w ,...,w beallmonomialsofdegreedinthefreealgebrakhx ,...,x i. Since
1 m 1 n
it is a linear basis in the dth homogeneous component of khx ,...,x i, we can
1 n
uniquely express the above polynomials up v as a linear combinations of w :
i l
m
up v = λl w .
i u,v,i l
Xl=1
The dimension of I is exactly the rank of the rectangularmatrix Λ={λl },
d u,v,i
whose rows of length m are labeled by the triples (u,v,i), where i = 1¯,r and
u,v are monomials in x ,...,x satisfying degu+degv =d−2.
1 n
Obviouslyλl arelinearfunctionsofthecoefficientsαk,l ofthepolynomials
u,v,i i
p . ConditionthatthedimensionofA isminimalisequivalenttothecondition
i d
that dimI = rkΛ is maximal. Denote the maximal rank of Λ by D. Thus,
d
the dimension of A is minimal if and only if there is a non-zero minor of
d
the matrix Λ of the size D. The family of the minors of Λ of the size D is
a finite family of polynomials P on the coefficients αk,l and some of these
l i
polynomialsarenon-zero. Thismeans thatthe setofA∈A(n,r) withminimal
dim A corresponds to the complement of the union of the sets of zeros of
d
finitely many non-zero polynomials. Any such set is a non-empty Zarisskii
open set and its complement has zero Lebesgue measure. The set of algebras
A∈A(n,r)satisfyingH =Hn,r isthenacountableintersectionofnon-empty
A min
Zarisskii open sets and therefore its complement has zero Lebesgue measure as
a countable union of sets with the Lebesgue measure zero. This completes the
proof of the proposition.
Remark Let us mention that in case when the minimal series Hn,r is
min
finite, the countable union from the proofof the Proposition2.1 is in fact finite
and the equality H = Hn,r is satisfied for A ∈ A(n,r) in general position in
A min
the Zarisskii sense as well and over an arbitrary field.
2.2 The Anick conjecture holds for n 6 7
Now we are back to arbitrary basic field k of char 0. We consider the question
whether the minimalseriesis finite forthe caser =n(n−1)/2. Itwasraisedin
thepaperofD.Anick[1],wherealowerboundfortheHilbertseriesforalgebras
from A(n,n(n−1)) was discovered. It was established that
2
−1
Hn,n(n−1)/2 > 1−nt+ n(n−1)t2 ,
min (cid:12)(cid:12)(cid:18) 2 (cid:19) (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where > is a componentwise ine(cid:12)quality, i.e. it holds if eac(cid:12)h coefficient of the
first series is greater or equal then the corresponding coefficient of the second
oneand|f(t)|standsforthepositivepartoftheseriesf ∈k[[t]]. Moreprecisely,
if f(t) = a +a t+a t2..., then |f(t)| = b +b t+b t2..., where b = a
0 1 2 0 1 2 m m
for m ∈ {i|a > 0 ∀j 6 i} and b = 0 otherwise. There was a question raised
j m
whether this lower bound is attained.
Since we know from the theorem 2.1 that the algebras of A(n,r) in general
position have minimal Hilbert series, to prove that this estimate attained it
4
would be enough to be able to write down generic coefficients of the relations
and calculate the Hilbert series.
Example 1. The algebra A over the field k =Z given by the relations
17
ac+2ba+9b2+3ca+9cb+8c2,
A=kha,b,ci 3ab+5ac+7ba+b2+8bc+4ca+cb+2c2,
. 10a2+2ab+11ac+2ba+8b2+4bc+9ca+7cb+5c2
has the Hilbert series H =1+3t+6t2+9t3+9t4 =|(1−3t+3t2)−1|. 1
A
By this method we are able to confirm Anick’s conjecture for small number
of generators.
Proposition 2.2 The lower bound for the Hilbert series of an algebra A ∈
A(n,n(n−1))overafieldkofchar0givenby 1−nt+ n(n−1)t2 −1 isachieved
2 (cid:12) 2 (cid:12)
(cid:12)(cid:16) (cid:17) (cid:12)
for n67. (cid:12) (cid:12)
(cid:12) (cid:12)
Proof. In Example 1 we have been calculating over the field k of charac-
teristicp=17. Sincethe series|(1−3t+3t2)−1|whichisknowntobethe lower
bound coincide with the resultofour calculations,we actually haveshownthat
for any term of the series, rank of matrix Λ = {λl } formed as above in the
u,v,i
proof of the proposition 2.1, with λl ∈Z is maximal. We now can see that
u,v,i p
rank of the same matrix considered over k is also maximal. Indeed, the reduc-
tion from Z to Z we have because rkM(Z )6rkM(Z). Then since chark =0,
p p
we have Z embedded into k and the same matrix has maximal rank overk. So,
if over the field Z for some p, the rank is maximal, then it is maximal over k.
p
Similarly, we have got an examples of algebras with the Hilbert series 1+
nt+ n(n+1)t2+n2t3+n2(n− (n2−1))t4 for n=4 and 1+nt+ n(n+1)t2+n2t3
2 2 2
for n=5,6,7,which are coincide with the series |(1+nt+ n(n−1)t2)−1| for these
2
values of n.
3 RIT algebras and maps of the finite set
3.1 The class of RIT algebras
HereweconsiderasubclassRoftheaboveclassofquadraticalgebrasA(n,n(n−1)).
2
TheclassRofRIT(relativisticinternaltime)algebrasconsistsofhomogeneous
finitely generated quadratic algebras given by relations of type (1).
ItturnedoutthatifrelationsformaGr¨obnerbasis,thealgebrasfromRare
so called ”geometric rings”, more precisely they are Auslander regular, Cohen–
Macaulay, gldimR = GKdimR = n (number of generators) for them. We
have proved this in [9] using combinatorial techniques related to the notion of
an I-type algebra introduced by J.Tate and M.Van den Bergh in [16]. These
arguments appeared due to inspiring question of M.Van den Bergh on whether
RIT algebras obey these properties. It becomes clear later on that Auslander
regularityet.al. couldalso be provedwithout employingthe I-type property,in
more general context, using arguments involving associated graded structures
with respect to appropriate grading (see [13], [12]).
The origin of the class of RIT algebras could be described in such a way.
The Lie algebra RIT was introduced in [4] as a modification of the Poincare
algebra P = L +U. Here L = O(3,1) is a Lorenz algebra but the space
4 4 4
1ThecomputationsweredoneusingbunchofprogramsGRAAL(GradedAlgebras)written
inUljanovskbyA.KondratyevundertheguideofA.Verevkin.
5
U changed by addition of a new variable T (related to the relativistic internal
time) to the set of initial variables. The corresponding commutation relations
containing T werederived. Taking an envelopingalgebraof P andconsidering
4
theassociatedgradedalgebraweobtaintheassociativeRITalgebrawhichgives
rise to the class under consideration.
Let we mention that the simplest algebra from this class R = R =
1,1
khx,yi (xy −yx−y2) is one of the two Auslander regular algebras of global
dimens(cid:14)ion2,thesecondoneistheusualquantumplanekhx,yi (xy−qyx)(this
follows from the Artin, Shelter classification [5]). We have bee(cid:14)n studying finite
dimensional representations of it in [10].
3.2 Condition on maps and Gr¨obner basis
The presentation (1) of the algebra gives us a set of maps σ : N →N defined
i
as
σ (j)=f(i,j) ∀j ∈N ={1,...,n}, i∈M ={1,...,m}.
i
We are interested to relate the properties of algebras to the properties of
these maps.
In particular, we will clarify combinatorialconditions on the associated col-
ored graph (=set of maps) which mean that relations form a Gro¨bner basis.
This gives at the same time a condition equivalent to the maximality of the
Hilbert series. Here we mean a lexicographicalorder on the series.
To speak about the Gr¨obner basis we have to fix an ordering on the set of
variables,letx >y foranyi,j andx >x fori>j,y >y fori>j. Onthe
i j i j i j
monomialsofvariablesx ,y theorderissupposedtobedegree-lexicographical.
i j
Then we rewrite relations (1) in the form
[x ,x ]=0, ∀i>j
i j
F = [yi,yj]=0, ∀i>j (2)
[x ,y ]=|y y |, ∀i,j
i j f(i,j) j
Here |y y | stands for the normal form of this monomial, i.e.
f(i,j) j
y y if f(i,j)6j
|y y |= f(i,j) j (3)
f(i,j) j (cid:26) yjyf(i,j) if f(i,j)>j
Theorem 3.1 The relations of the type (2) form a reduced Gro¨bner basis if
and only if the function f(i,j) defines the set of actions σ with the following
i
property. Foranypairofmapsσ ,σ , k >ioneofthetwoconditions issatisfied
i k
in each point j ∈N:
either (1). σ (j)=σ (j) and σ σ (j)=σ σ (j)
k i i k k i
or (2). σ (j)=σ σ (j) and σ (j)=σ σ (j).
k k i i i k
Let we mention that the second condition implies kind of strong version of
braid type relations on σ : σ =σ σ σ and σ =σ σ σ for k >i.
i k k i k i i k i
Proof. TheproofisadirectapplicationofGr¨obnerbasestechnique dueto
Buchberger [7] and Bergman [6].
To find out that relations form a Gr¨obner basis we have to check that all
ambiguities are solvable. Possible ambiguities in our case are of four types:
1. x x x , i>j >k,
i j k
2. y y y , i>j >k,
i j k
3. x y y , ∀i,j >l,
i j l
4. x x y , ∀j,l>i.
l i j
6
The first two types are trivially solvable. Ambiguities of the type three are
also solvable:
y x y +y y y ←− x y y −→ x y y
j i l (i,j) j l i j l i l j
↓ ↓
y y x +y y y y x y +y y y
j l i j (i,l) l l i j (i,l) l j
↓
y y x +y y y
l j i l (i,j) j
Ambiguities of the type four are solvable if and only if the following two-
element (non-ordered) sets are coincide:
{f(i,j);f(l,f(i,j))}={f(i,f(l,j));f(l,j)},
for any j and l>i.
Indeed:
x x y ←− x x y −→ x y x +x |y y |
i l j l i j l j i l (i,j) j
↓ ↓
y x x +y y x +y x y +
j l i (l,j) j i (i,j) l j
x y x +x |y y |
i j l i (l,j) j +y y y
(l,(i,j)) (i,j) j
↓ ↓
y x x +y y x +y x y +
j i l (i,j) j l (l,j) i j
y y x +y y y
+y y y (i,j) j l (i,j) (l,j) j
(i,(l,j)) (l,j) j
↓
y y x +y y y
(l,j) j i (l,j) (i,j) j
Insomeplacesabovewewriteforexample|y y |insteadofy y . These
(i,j) j (i,j) j
are those places where order on y essential for the future reductions (namely
i
we have some x before y ). We actually had to check all possibilities for the
j i
pairs |y y |, |y y | appearing at the above sequences of reductions and all
(l,j) j (i,j) j
of them via different cancelations gave the same result.
The coincidence of above mentioned sets means in the language of maps σ
i
that
{σ (j);σ σ (j)}={σ σ (j);σ (j)},
i l i i l l
for any j and l>i.
These sets are coincide if and only if for any fixed l > i in each point j
we have either σ (j) = σ (j) and σ σ (j) = σ σ (j) or σ (j) = σ σ (j) and
i l l i i l i i l
σ σ (j)=σ (j). By this we are done.
l i l
Itisofcourseverynaturalandimportantquestion,whengivenpresentation
ofanalgebraformaGr¨obnerbasis. InRITcasetheseconditionstakeaspecific
shape of description of defining maps σ , obtained above. Conditions for that
i
were formulated also for example for the class of G-algebras in [11] under the
name of non–degeneracy condition. Now we turn to more difficult matter of
clarifying a precise combinatorial structure of maps obeying conditions of the
theorem 3.1. As a first step we consider few particular cases, which we will use
later on to prove the general fact.
3.3 Representations of the semigroup hx |x = x x i
i i i j
Here we consider the case when all elements of N obey conditions (2) from the
theorem 3.1. That is we have σ (j) = σ σ (j) and σ (j) = σ σ (j) for any
k k i i i k
k >i,j ∈N. Thismeansthatσ sformarepresentationbyactionsonthe finite
i
set of the semigroupΩ=hx |x =x x ,16i6=j 6mi. From these relations it
i i i j
7
followsthatallx areidempotents. Reductionofthefirstsubwordx x inx x x
i i j i j i
gives x x x = x x , of the second one: x x x = x x , but then x x = x .
i j i i i i j i i j i j i
Hence we could also write relations just like Ω = hx |x = x x ,i,j = 1,¯mi,
i i i j
without the conditioni6=j. Note, that this semigroupconsists in factof m+1
elements: any word in this semigroup is equal to its first letter.
What is the structure of maps which form representations then?
Theorem 3.2 Any representation of the semigroup Ω=hx |x x =x i has the
i i j i
following structure. The set of representation N is decomposed into a disjoint
union of subsets. In each of them there are m fixed points (not necessarily
different), such that the maps σ , k =1,¯m send the entire subset to the kth of
k
these points.
Proof. Let {σ }m be a representation of the semigroup Ω on the set
k k=1
N. That is, σ σ = σ for any 1 6 j,k 6 m. First note that since σ are
j k j k
idempotents, they are identity on their images: σ (m)=m for each m∈R =
k k
Imσ . Define the equivalence relation on N corresponding to σ : m ∼
k 1 1 σ1
m if σ (m ) = σ (m ). Then N splits into the union of equivalence classes
2 1 1 1 2
O ,...,O , where each class contains a unique element s from R = Imσ ,
s1 sk j j j
so we can enumerate these classes by these elements. Consider the restriction
of the maps σ to an arbitrary class O . Since σ σ = σ , each σ leaves
k sj j k j k
the set O invariant. Indeed, σ (σ (r)) = σ (r) = s for each r ∈ O and
sj j k j j sj
therefore σ (r) ∈ O . Moreover, since σ σ = σ , the set σ (O ) consists
k sj k j k k sj
of one element σ (s ). Indeed, σ (r) = σ (σ (r)) = σ (s ) for each r ∈ O .
k j k k j k j sj
Hence the structure of these maps is the following: the set N is decomposed
into a disjoint union of subsets, in each of which m points are chosen and the
maps σ map the entire subset to the kth of these points. Some of these points
k
could coincide.
Obviously,theotherwayaround,ifonetakeanysetofmaps{σi}i=1,¯m with
the described structure, then they form a representation of the semigroup Ω,
that is satisfy the relations σ σ =σ ,∀k,i=1,¯m.
k i k
Thusthereexists1-1correspondencebetweenrepresentationsofΩonafinite
set and maps described in the theorem 3.2.
Let we mention that the same is true for representations on an infinite set,
our arguments work there without any change.
It is natural to ask when there exists a faithful representation.
Corollary 3.3 For any n>m there exists a faithful representation of Ω on the
set of size n.
Proof. TheimageofthesemigroupΩinthesetofmapsconsistsjustofthe
maps σ ,...,σ ,which areimagesofthe generatorsx ,...,x ofthe semigroup.
1 m 1 m
This follows from the relations. Hence if we can just take m different maps of
the required nature, then they form a faithful representation. It is certainly
possible if n > m: take σ (j) = r ,...,σ (j) = r ,r ∈ N. For different r we
1 1 m m i i
get different maps.
It is possible to find faithful representations of smaller dimension. For ex-
ample take a subsets from theorem 3.2 of size 3. Namely, let m = 3d and our
representation set consists of the pairs N = {(k,ε)|k = 0,...,d−1,ε = 0,1,2}.
Maps define as follows: σ (k,ε)= (k,ε (i)), where ε (i) is an ith coefficient in
i k k
presentation of i in base 3: i=ε0(i)+3ε1(i)+...+3d−1εd−1(i). We have then
a faithful representation on the set N of size 3⌈log m⌉.
3
Itisnotdifficulttoshowthatasymptoticallythisstrategygivesthebestpos-
sible result, so asymptotically minimal size of faithful representationis 3log m.
3
8
3.4 Combinatorial description of maps corresponding to
maximal Hilbert series
Here we give a combinatorial description of the maps σ , i = 1,2 satisfying
i
condition(1)or(2)astheyappearinthetheorem3.1above,thatisthosemaps
which define an algebra with maximal Hilbert series.
We also prove as a consequence that if Hilbert series is maximal then all
maps σ , i=1,¯m coming from defining relations of arbitrary RIT algebrahave
i
pairwise 2-isomorphic graphs.
Consider maps σ and σ . Suppose they obey conditions described in theo-
i k
rem 3.1. Let us define the set Y as a set of elements j ∈ N where σ and σ
0 i k
coincide:
Y ={j ∈N|σ (j)=σ (j)}.
0 i k
Lemma 3.4 Y is invariant under the action of both σ and σ
0 i k
Proof. Let j be from Y . If in point j σ and σ obey condition (1) form
0 i k
theorem 3.1, then σ (j) ∈ Y and σ (j) ∈ Y due to the second part of (1).
i 0 k 0
Indeed, values of maps σ and σ on their images should coincide, hence these
i k
imagesareagaininY . Supposeinpointj condition(2)isfulfilled. Sincej ∈Y
0 0
we have σ (j) = σ (j) = r, but due to (2): r = σ (j) = σ σ (j) = σ (r) and
i k k k i k
r =σ (j)=σ σ (j)=σ (r). Which means that actually (1) holds also for this
i i k i
point j. So for each point j ∈ Y condition (1) is satisfied. From this it easy
0
follows that σ (Y )⊂Y and σ (Y )⊂Y .
i 0 0 k 0 0
Letnowconsideranelementj ∈/ Y anditsimagesj =σ (j)andj =σ (j).
0 1 i 2 k
Sinceimagesofσ andσ aredifferentinj,condition(1)couldnotholdsinthis
i k
point, thus we have condition (2) there. This gives us: j =σ (j)=σ σ (j)=
2 k k i
σ (j ) and j = σ (j) = σ σ (j) = σ (j ). So we have that σ maps j to j
k 1 1 i i k i 2 k 1 2
and σ maps j to j .
i 2 1
Using this information let us clarify how the element from outside Y could
0
get to Y . Suppose j ∈/ Y but j = σ (j) ∈ Y . Then σ (j ) = σ (j ) = j .
0 0 1 i 0 k 1 i 1 2
Thismeansthatnotonlyj goestoj underσ ,butalsothe otherwayaround,
2 1 i
σ maps j to j . On Y condition (1) from the theorem 3.1 always holds and
i 1 2 0
j ∈ Y therefore for j which is image of j under σ we have σ (j ) = σ (j )
1 0 2 1 i i 2 k 2
(due to the second part of condition (1) in point j ). Since j =σ (j )∈Y , j
1 1 i 2 0 2
is also in Y .
0
We have proved
Lemma 3.5 Let j ∈/ Y but σ (j)∈Y . Then images of j under σ and σ both
0 i 0 i k
are in Y and σ as well as σ maps them to each other.
0 i k
Lemma 3.6 If j ∈/ Y but j =σ (j)∈Y , then there is no such element from
0 1 i 0
N, which has an image j under σ or σ .
i k
Proof. Suppose there exists m ∈ N, such that say σ (m) = j. Obviously
i
m∈/ Y ,sinceY isinvariantandthenj shouldbeinY ,butitisnot. Forpoints
0 0 0
whicharenotinY condition(1)couldnotholds,thus wehavecondition(2)in
0
m. This leads to the following contradiction: σ (m) = σ σ (m) = σ (j) = j ,
k k i k 2
σ (m)=σ σ (m)=σ (j )=j hence σ (m)=j and σ (j)=j , but j can not
i i k i 2 1 i i 1 1
be equal to j just because one is from Y and another is not.
0
We are now in a position to define a bigger set Y˜ :
0
Y˜ =Y ∪{j ∈N|σ (j)∈Y },
0 0 i 0
which satisfy the following nice property.
9
Lemma 3.7 The set N splits on twodisjoint subsets which areinvariant under
σ and σ :
k i
N =Y˜ ⊕P
0
where P = N\Y˜ . Moreover the structure of maps on P is precisely as it was
0
describedinthetheorem3.2: P isadisjointunionofsubsetsonwhichtwopoints
are picked such that σ maps entire subset to one of them and σ to another.
i k
Proof. The fact that σ and σ preserve Y was proved above, invariance
i k 0
of Y˜ then follows from its definition. Invariance of complement P of Y˜ comes
0 0
from the statement of lemma 3.6.
Let we define one more subset: Z =Y \ σ (Y˜ \Y ).
0 j 0 0
j=Si,k
Abovelemmataallowustogivethefollowingprecisedescriptionofmapsσ ,
i
σ corresponding to the maximal Hilbert series.
k
Theorem 3.8 Algebra R∈R(2,n) has a maximal Hilbert series if and only if
maps σ , σ coming from defining relations has the following structure.
i k
The set N is a disjoint union of invariant under both maps subsets P and
Y˜ : N =P ⊕Y˜ .
0 0
The action on P is the following: P is a disjoint union of P , in each of
i
them two points (not necessary different) are fixed, such that σ maps entire P
i i
to one of them and σ to another.
j
The map on the other disjoint component Y˜ is the following: there are three
0
subsetsZ ⊆Y0 ⊆Y˜0. ThesetY0\Z isadisjointunionofpairs{j1(l),j2(l)}l∈Σ, (Σ
is afinitesetofindexes), suchthat ∀j ∈Y˜ \Y ∃l∈Σ:σ (j)=j(l),σ (j)=j(l)
0 0 i 1 k 2
and σ (j(l)) = j(l), σ (j(l)) = j(l). Values of σ and σ on Z are in Y and
i,k 1 2 i,k 2 1 i k 0
coincide.
Let we say that two graphs are 2-isomorphic if their images under gluing
pairs of vertices of common cycles of length two are isomorphic.
As a consequence of the above theorem we get the following
Corollary 3.9 If an algebra R ∈ R(m,n) has a maximal Hilbert series, then
all graphs of maps σ coming from the defining relations of R are pairwise 2-
i
isomorphic.
Proof. Note that the feature of condition on σ formulated in theorem 3.1
i
isthatitissatisfiedforanarbitraryset{σi}i=1,¯m ifandonlyifitissatisfiedfor
anypairσ ,σ , i<k. Sowecanapplytheorem3.8foranyfixedpairσ ,σ . It
i k i k
is clear from the above description that graphs of maps σ and σ are the same
i k
or isomorphic with the isomorphism defined by permuting of some pairs j ,j .
1 2
Except one possible situation when for some pair j ,j ∈ Y \Z which consists
1 2 0
ofimages ofj ∈Y˜ \Y we havea pointl ∈Z suchthat σ (l)=σ (l)=j . This
0 0 i k 1
givesapossibilityforgraphsofσ andσ tobenonisomorphic. Weexcludethis
i k
possibility by gluing vertices of common cycles of length two (j and j ) in this
1 2
two graphs, so after such an operation graphs become isomorphic.
Corollary 3.10 Combinatorial conditions from the theorem 3.8 on σ ,σ are
i k
equivalent to the following properties of algebra R∈R(m,n):
(i) R has a quadratic Gro¨bner basis;
(ii) R has a lexicographically maximal in R(m,n) Hilbert series;
(iii) R is a PBW algebra (that is, have a series H = 1 ),
R (1−t)(m+n)
and imply the following properties of algebra:
(iv) R is Koszul;
10
