A Mathematical Journal
Vol. 7, No 2, (1-20). August 2005.

Gröbner and diagonal bases in
Orlik-Solomon type algebras

Raul Cordovil 1

Departamento de Matemática, Instituto Superior Técnico
Av. Rovisco Pais. 1049-001 Lisboa - Portugal

cordovil@math.ist.utl.pt

David Forge
Laboratoire de Recherche en Informatique. Batiment 490 Universite Paris Sud

91405 Orsay Cedex - France
forge@lri.fr

ABSTRACT
The Orlik-Solomon algebra of a matroid M is the quotient of the exterior

algebra on the points by the ideal �(M) generated by the boundaries of the
circuits of the matroid. There is an isomorphism between the Orlik-Solomon al-
gebra of a complex matroid and the cohomology of the complement of a complex
arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon
algebras, called χ-algebras, are considered. These new algebras include, apart
from the Orlik-Solomon algebras, the Orlik-Solomon-Terao algebra of a set of
vectors and the Cordovil algebra of an oriented matroid. To encode an important
property of the “no broken circuit bases” of the Orlik-Solomon-Terao algebras,
András Szenes has introduced a particular type of bases, the so called “diagonal
bases”. This notion extends naturally to the χ-algebras. We give a survey of the
results obtained by the authors concerning the construction of Gröbner bases of
�χ(M) and diagonal bases of Orlik-Solomon type algebras and we present the
combinatorial analogue of an “iterative residue formula” introduced by Szenes.

1The first author’s research was supported in part by FCT (Portugal) through program POCTI
and the project SAPIENS/36563/99.



2 Raul Cordovil and David Forge
7, 2(2005)

RESUMEN
El álgebra de Orlik-Solomon de una matroide M es el cuociente del álgebra

exterior en los puntos por el ideal �(M) generado por los acotamientos de los cir-
cuitos de la matroide. Existe un isomorfismo entre el álgebra de Orlik-Solomon de
una matroide compleja y la cohomoloǵıa del complemento de un arreglo complejo
de hiperplanos. En este art́ıculo se considera una generalización de las algebras de
Orlik-Solomon, llamadas χ-algebras. Estas nuevas álgebras incluyen, además de
las álgebras de Orlik-Solomon, el álgebra de Orlik-Solomon-Terao de un conjunto
de vectores y el álgebra de Cordovil de una matroid orientada. Para recalcar una
importante propiedad de las ”bases de circuitos no quebrados” de las álgebras de
Orlik-Solomon-Terao, András Szenes ha introducido un particular tipo de bases,
llamadas ”bases diagonales”. Este concepto se extiende naturalemente a la χ-
algebras. Damos una mirada a los resultados obtenidos por los autores referentes
a la construcción de las bases de Gröbner de �χ(M) y bases diagonales de los
tipos de algebras de Orlik-Solomon, y presentamos el análogo combinatorio de
una ”fórmula de residuos iterativa” introducida por Szenes.

Key words and phrases: arrangement of hyperplanes, broken circuit,
cohomology algebra, matroid, oriented matroid,
Orlik-Solomon algebra, Gröbner bases.

Math. Subj. Class.: Primary: 05B35, 52C35; Secondary: 14F40.

1 Introduction

Let M = M([n]) be a matroid on the ground set [n] := {1, 2, . . . ,n}. The Orlik-
Solomon algebra of a matroid M is the quotient of the exterior algebra on the points
by the ideal �(M) generated by the boundaries of the circuits of M. The isomor-
phism between the Orlik-Solomon algebra of complex matroid and the cohomology
of the complement of a complex arrangement of hyperplanes was established in [12].
The Orlik-Solomon algebras have been then intensively studied. A general reference
on hyperplane arrangements and Orlik-Solomon algebras is [14]. Descriptions of de-
velopments from the early 1980’s to the end of 1999, together with the contributions
of many authors, can be found in [9, 21].

In this article a generalization of the Orlik-Solomon algebras, called χ-algebra, is
considered. These new algebras include, apart from the Orlik-Solomon algebras, the
Orlik-Solomon-Terao algebra of a set of vectors [15] and the Cordovil algebra of an
oriented matroid [7]. We will survey recent results concerning this family of Orlik-
Solomon type algebras (see [8, 10, 11]). In this introduction, we will recall the origin
of the Orlik-Solomon algebra and we will develop the different notions used in the
next sections like matroids and oriented matroids, the Orlik-Solomon algebra and its
generalizations, its diagonal bases and the Gröbner bases of the defining ideal.

Let V be a vector space of dimension d over some field K. A (central) arrangement
(of hyperplanes) in V, AK = {H1, . . . ,Hn}, is a finite listed set of codimension one



7, 2(2005)
Gröbner and diagonal bases in Orlik-Solomon type algebras 3

vector subspaces. Given an arrangementAK we always suppose fixed a family of linear
forms

{
θHi ∈ V ∗ : Hi ∈AK, Ker(θHi ) = Hi

}
, where V ∗ denotes the dual space of V .

Let L(AK) be the intersection lattice of AK: i.e., the set of intersections of hyperplanes
in AK, partially ordered by reverse inclusion. There is a matroid M(AK) on the
ground set [n] determined by AK: a subset D ⊆ [n] is a dependent set of M(AK) iff
there are scalars ζi ∈ K, i ∈ D, not all nulls, such that

∑
i∈D ζiθHi = 0. A circuit is a

minimal dependent set with respect to inclusion. If K is an ordered field an additional
structure is obtained: to every circuit C,

∑
i∈C ζiθHi = 0, we associate a partition

(determined up to a factor ±1) C+ := {i ∈ C : ζi > 0},C− := {i ∈ C : ζi < 0}.
With this new structure M(AK) is said a (realizable) oriented matroid and denoted
by M(AK). Oriented matroids on a ground set [n], denoted M([n]), are a very
natural mathematical concept and can be seen as the theory of generalized hyperplane
arrangements, see [3].

Set M(AK) := V \
⋃

H∈AK H. The manifold M(AC) plays an important role in
the Aomoto-Gelfand theory of multidimensional hypergeometric functions (see [16]
for a recent introduction from the point of view of arrangement theory). Let K
be a commutative ring. In [12, 13, 14] the determination of the cohomology K-
algebra H∗

(
M(AC); K

)
from the matroid M(AC) is accomplished by first defining

the Orlik-Solomon K-algebra OS(AC) in terms of generators and relators which de-
pends only on the matroid M(AC), and then by showing that this algebra is iso-
morphic to H∗

(
M(AC); K

)
. Aomoto suggested the study of the (graded) K-vector

space AO(AK), generated by the basis {Q(BI)−1}, where I is an independent set of
M(AK), BI := {Hi ∈AK : i ∈ I}, and Q(BI) =

∏
i∈I θHi denotes the corresponding

defining polynomial. To answer a conjecture of Aomoto, Orlik and Terao have in-
troduced in [15] a commutative K-algebra, OT(AK), called the Orlik-Solomon-Terao
algebra. The algebra OT(AK) is isomorphic to AO(AK) as a graded K-vector space
in terms of the equations {θH : H ∈ AK}. A “combinatorial analogue” of the alge-
bra of Orlik-Solomon-Terao was introduced in [7]: to every oriented matroid M was
associated a commutative Z-algebra, denoted by A(M) and called the Cordovil al-
gebra. The χ-algebras generalizes the three just mentioned algebras: Orlik-Solomon,
Orlik-Solomon-Terao and the Cordovil algebras, see [11] or Example 2.4 below.

In section two we will give the definition of a χ-algebra and recall the principal
examples. In general a χ-algebra, denoted Aχ(M), is defined as the quotient of some
kind of a finite K-algebra A by an ideal �χ(M) of A whose generators are defined from
the circuits of M and are depending of the map χ, see Definition 2.2. In particular
the first important result is that like for the original Orlik-Solomon algebra we get
nbc-bases of the χ-algebra (as a module) from the “no broken circuit” sets of the
matroid and corresponding basis for the ideal �χ(M).

In section three, we construct the reduced Gröbner basis of the ideal �χ(M) for
any term order ≺ on the set of the monomials T(A) of the algebra A. This result
gives as a corollary a universal Gröbner basis (a Gröbner basis who works for every
term order) which is shown to be minimal. Finally we remark that the nbc-bases
are in some sense the bases corresponding to the Gröbner bases for the different term
orders.



4 Raul Cordovil and David Forge
7, 2(2005)

In section four, following Szenes [17], we define a particular type of basis of Aχ,
the so called “diagonal basis”, see Definition 4.7. The nbc-bases are an important
examples of diagonal bases. We construct the dual bases of these bases, see Theo-
rem 4.8. Our definitions make also use of an “iterative residue formula” based on the
matroidal operation of contraction, see Equation (4.6). This formula can be seen as
the combinatorial analogue of an “iterative residue formula” introduced by Szenes,
[17]. As applications we deduce nice formulas to express a pure element in a diagonal
basis. We prove also that the χ-algebras verify a splitting short exact sequence, see
Theorem 4.4. This theorem generalizes for the χ-algebras previous similar theorems
of [7, 14].

We use [19, 20] as a general reference in matroid theory. We refer to [3] and [14]
for good sources of the theory of oriented matroids and arrangements of hyperplanes,
respectively.

2 χ- algebras

Let IND�(M) ⊆
(
[n]
�

)
[resp. DEP�(M) ⊆

(
[n]
�

)
] be the family of independent [resp.

dependents] sets of cardinality � of the matroid M and set

IND(M) :=
⋃
�∈N

IND�(M),

DEP(M) :=
⋃
�∈N

DEP�(M).

We denote by C = C(M) the set of circuits of M. For shortening of the notation
the singleton set {x} is denoted by x. When the smallest element α of a circuit C,
|C| > 1, is deleted, the remaining set, C\α, is said to be a broken circuit. (Note that
our definition is slightly different from the standard one. In the standard definition
C\α can be empty.) A no broken circuit set of a matroid M is an independent subset
of [n] which does not contain any broken circuit. Let NBC�(M) ⊆

(
[n]
�

)
be the set of

the no broken circuit sets of cardinal � of M and set
NBC(M) :=

⋃
�∈N

NBC�(M).

Let L(M) be the lattice of flats of M.
(
We remark that the lattice map φ :

L(AK) → L(M(AK)), determined by the one-to-one correspondence φ′ : Hi ←→{i},
i = 1, . . . ,n, is a lattice isomorphism.

)
For an independent set I ∈ IND(M), let c�(I)

be the closure of I in M.
For every permutation σ ∈ Sm, let Xσ be the ordered set

Xσ := iσ(1) ≺ ···≺ iσ(m) = (iσ(1), iσ(2), . . . , iσ(m)).
When necessary we also see the set X = {i1, . . . , im}, as the ordered set

Xid = (i1, . . . , im).



7, 2(2005)
Gröbner and diagonal bases in Orlik-Solomon type algebras 5

Set Xσ \x := (iσ(1), . . . , x̂, . . . , iσ(m)). If Y β = (jβ(1), . . . ,jβ(m′)) and X ∩Y = ∅, let
Xσ ◦Y β be the concatenation of Xσ and Y β , i.e., the ordered set

Xσ ◦Y β := (iσ(1), . . . , iσ(m),jβ(1), . . . ,jβ(m′)).
Definition 2.1 Let χ be a mapping χ : 2[n] → K. Let us also define χ for ordered
sets by χ(Xσ) = sgn (σ)χ(X), where sgn (σ) denotes the sign of the permutation σ.
Fix a set E = {e1, . . . ,en}. Let A := K ⊕ A1 ⊕···⊕ An be the graded algebra over
the field K generated by the elements 1,e1, . . . ,en and satisfying the relations:

◦ 1ei = ei1 = ei, for all ei ∈ E,
◦ e2i = 0, for all ei ∈ E,
◦ ej ·ei = βi,jei ·ej with βi,j ∈ K∗ for all i < j.
By definition the χ-boundary of an element eX ∈ A, X �= ∅, is given by the formula

∂eX :=
p=m∑
p=1

(−1)pχ(X \ ip)eX\ip.

We set ∂ei = 1, for all ei ∈ E. We extend ∂ to the K-algebra A by linearity.
Let X = (i1, i2, . . . , im). In the sequel we will denote by eX the (pure) element of the
K-algebra A,

eX := ei1 ·ei2 · · · · ·eim.
By convention we set e∅ := 1. Both the exterior K-algebra,

∧
E, (take βi,j = −1) and

the polynomial algebra K[e1, . . . ,en]/〈e2i〉 with squares zero (take βi,j = 1) considered
in [7, 15], are such K-algebras A and will be the only ones to be used in the examples.
It is clear that for any x �∈ X,

±∂eX∪x = (−1)m+1χ(X)eX +
p=m∑
p=1

(−1)pχ(X \ ip ◦x)eX\ip∪x.

From the equality χ(Xσ) = sgn (σ)χ(X), it is easy to see that for σ ∈ S|X| we have

∂eX = sgn (σ)
p=m∑
p=1

(−1)pχ(Xσ \ iσ(p))eX\iσ(p).

Given an independent set I, an element a ∈ c�(I) \ I is said active in I if a is the
minimal element of the unique circuit contained in I ∪ a. We say that a subset
U ⊆ [n] is a unidependent set of M([n]) if it contains a unique circuit, denoted C(U).
Note that U is unidependent iff rk(U) = |U| − 1. We say that a unidependent set
U is an inactive unidependent if min(C(U)) is the the minimal active element of
U \ min(C(U)). We will denote by UNI�(M) for the sets of inactive unidependent
sets of size � and set

UNI(M) :=
⋃
�∈N

UNI�(M).



6 Raul Cordovil and David Forge
7, 2(2005)

Let us remark that U is a unidependent set of M iff for some (or every) x ∈ U,
rk(x) �= 0, U \x is a unidependent set of M/x.
Definition 2.2 ([11]) Let χ be a mapping χ : 2[n] → K. Let �χ(M([n])) be the
(right) ideal of A generated by the χ-boundaries {∂eC : C ∈ C(M), |C| > 1} and the
set of the loops of M, {ei : {i} ∈ C(M)}. We say that Aχ(M) := A/�χ(M) is a
χ-algebra if χ satisfies the following two properties:

(2.2.1) χ(I) �= 0 if and only if I is independent.
(2.2.2) For any two unidependents U and U′ of M with U′ ⊆ U there is a scalar

ε
U,U′ ∈ K∗, such that ∂eU = εU,U′ (∂eU′ )eU\U′ .

Note that
{eC : C ∈ C(M)}⊆�χ(M([n])).

For every X ⊆ [n], we denote by [X]A or shortly by eX when no confusion will
result, the residue class in Aχ(M) determined by the element eX . Since �χ(M) is
a homogeneous ideal, Aχ(M) inherits a grading from A. More precisely we have
Aχ(M) = K ⊕ A1 ⊕···⊕ Ar, where A� = A�/A� ∩�χ(M) denotes the subspace of
Aχ(M) generated by the elements

{
[I]A : I ∈ IND�(M)

}
. Set

nbc � :=
{
[I]A : I ∈ NBC �(M)

}
and nbc :=

⋃
�=0

nbc �,

dep� :=
{
[D]A : D ∈ DEP�(M)

}
and dep :=

⋃
�=0

dep �,

uni� :=
{
[U]A : U ∈ UNI�(M)

}
and uni :=

⋃
�=0

uni �.

Remark 2.3 From (2.2.1) and (2.2.2) we conclude that �χ(M) has the basis dep∪
∂uni and that nbc :=

{
[I]A : I ∈ NBC(M)

}
is a basis of the vector space A =

Aχ(M). We also have that nbc� is a basis of the vector space A�. This fundamental
property was first discovered for the Orlik-Solomon algebras [14], and then also for
the other classical χ-algebras, see [7, 15] and the following example for more details.
Note also that this implies that [X]A �= 0 iff X is an independent set of M.
Example 2.4 Recall the three usual χ-algebras Aχ(M).
◦ Let A = ∧E be the exterior K-algebra (taking βi,j = −1). Setting χ(Iσ) =

sgn(σ) for every independent set I of a matroid M and every permutation
σ ∈ S|I|, we obtain the Orlik-Solomon algebra, OS(M).

◦ Let AK = {Hi : Hi = Ker(θi), i = 1, 2, . . . ,n} be an hyperplane arrangement
and M(AK) its associated matroid. For every flat F := {f1, . . . ,fk} ⊆ [n] of
M(AK) we choose a bases BF of the vector subspace of (Kd)∗ generated by
{θf1, . . . ,θfk}. By taking A = K[e1, . . . ,en]/〈e2i〉 the polynomial algebra with
squares null (taking βi,j = 1) and taking for any {i1, . . . , i�} = I ∈ IND�,
χ(I) = det(θi1, . . . ,θi� ), where the vectors are expressed in the basis Bc�(I), we
obtain the Orlik-Solomon-Terao algebra OT(AK), defined in [15].



7, 2(2005)
Gröbner and diagonal bases in Orlik-Solomon type algebras 7

◦ Let M([n]) be an oriented matroid. For every flat F of M([n]), we choose
(determined up to a factor ±1) a bases signature in the restriction of M([n]) to
F . We define a signature of the independents of an oriented matroid M([n]) as a
mapping, sgn : IND(M) →{±1}, where sgn (I) is equal to the basis signature
of I in the restriction of M([n]) to c�(I). By taking A = Q[e1, . . . ,en]/〈e2i〉 the
polynomial algebra over the rational field Q with squares zero (take βi,j = 1) and
taking χ(I) = sgn (I) (resp. χ(X) = 0) for every independent (resp. dependent)
set of the matroid, we obtain the algebra A(M)⊕Z Q, where A(M) denotes the
Cordovil Z-algebra defined in [7].

3 Gröbner bases of χ-ideals

For general details on Gröbner bases of an ideal, see [1, 2]. We begin by adapting
some definitions to our context. Consider the K-algebra A introduced in Definition 2.1.
Note that there are monomials eY ,eZ ∈ A, such that eY · eZ = 0. In the standard
case where A is replaced by the polynomial ring K[e1, . . . ,en], this is not possible.
So the the following definitions are slightly different from the standard corresponding
ones given in [1, 2]. Let M = M([n]) be a matroid, �χ(M) and Aχ(M) the χ-ideal
and χ-algebra as defined in the previous section. We will denote for shortness A(M)
for Aχ(M).

Definition 3.1 Let T = T(A) be the set of the monomials of the K-algebra A, i.e.,
T(A) := {eX : X = (ei1, . . . ,eim )}. A total ordering ≺ on the monomials T is said
a term order on T if e∅ = 1 is the minimal element and ≺ is compatible with the
multiplication in A, i.e.,

∀eX,eY ,eZ ∈ T, (eX ≺ eY )&(eX ·eZ �= 0)&(eY ·eZ �= 0) =⇒ eX ·eZ ≺ eY ·eZ.

Given a term order ≺ on T and a non-null polynomial f ∈ A, we may write

f = a1eX1 + a2eX2 + · · · + ameXm,

where ai ∈ K∗ and eXm ≺ ··· ≺ eX1 . We say that the aieXi [resp. eXi ] are the
terms [resp. monomials] of f. We say that lp≺(f) := eX1 [resp. lt≺(f) := a1eX1 ]
is the leading monomial [resp. leading term] of f (with respect to ≺). We also
define lp≺(0) = lt≺(0) = 0. Note that in general we have lp≺(hg) �= lp≺(h)lp≺(g),
contrarily to the cases considered in [1, 2].

Example 3.2 A permutation π ∈ Σn defines a linear reordering of the set [n]:
π−1(1) <π π−1(2) <π · · · <π π−1(n). Consider the ordering of the set E

eπ−1(1) ≺π eπ−1(2) ≺π · · · ≺π eπ−1(n).

The corresponding degree lexicographic ordering on the monomials T, also denoted
≺π, is a term order on T.



8 Raul Cordovil and David Forge
7, 2(2005)

For a subset S, S ⊆ A and a term order ≺ on T(A), we define the leading term
ideal of S, denoted Lt≺(S), as the ideal generated by the leading monomials of the
polynomial in S, i.e.,

Lt≺(S) := 〈lp≺(f) : f ∈ S〉.
In the remaining of this section we suppose that M([n]) is a loop free matroid.
Definition 3.3 Let M be a matroid. Let ≺ be a term order on T(A). Consider the
ideal �χ(M) of A A family G of non-null polynomials of the ideal �χ(M) is called a
Gröbner basis of the ideal �χ(M) with respect to ≺ iff

Lt≺(G) = Lt≺(�χ(M)).

The Gröbner basis G is called reduced if, for every element g ∈ G we have lt≺(g) =
lp≺(g), and for every two distinct elements g,g

′ ∈ G, no term of g′ is divisible by
lp≺(g). The Gröbner basis G is called a universal Gröbner basis if it is a Gröbner
basis with respect to all term orders on T(A) simultaneously. If U is a universal
Gröbner basis, minimal for inclusion with this property, we say that U is a minimal
universal Gröbner basis.

From Definition 3.3 we conclude:

Proposition 3.4 Let G≺ be a Gröbner basis of the ideal �χ(M) with respect to the
term order ≺ on T(A). Then

BG≺ :=
{
eX + �χ(M) : eX �∈ Lt≺(G)

}
is a basis of the module Aχ(M).
We say that the well determined basis BG≺ is the canonical basis of the χ-algebra
Aχ(M) for the Gröbner basis G of the ideal �χ(M), with respect to the term order
≺ on T(A).

Consider the partition T(A) = Ti(A)
⊎

Td(A) where:

Ti(A) :=
{
eI : I ∈ IND(M)

}
and Td(A) :=

{
eD : D ∈ DEP(M)

}
.

Let K[Ti] and K[Td] be the K-vector subspaces of A generated by the basis Ti and Td,
respectively. So A = K[Ti] ⊕ K[Td]. We also see the set K[Td] ⊆�χ(M) as the ideal
of A generated by the set of monomials {eC : C ∈ C(M)}. Let pi : A → K[Ti] be the
first projection. We define the term orders on the set of monomials Ti in a similar
way to the corresponding definition on T. It is clear that the restriction of every term
order of T to the subset Ti is also a term order on Ti. We can also add to K[Ti] a
structure of K-algebra with the product 
 : K[Ti]×K[Ti] → K[Ti], determined by the
equalities

eX 
 eX′ = pi(eXeX′ ) for all X,X
′ ∈�χ(M).

Note that if eX 
 eX′ �= 0, then eX 
 eX′ = eXeX′ . We remember that eXeX′ �= 0 iff
X ∩X′ = ∅ and X ∪X′ ∈ IND(M). So �χi (M) := pi

(
�χ(M)

)
is an ideal of K[Ti].



7, 2(2005)
Gröbner and diagonal bases in Orlik-Solomon type algebras 9

Proposition 3.5 Let ≺ be a term order on T(A). Then the leading term ideals of
A, Lt≺(pi(�χ(M))) and Lt≺(�χ(M)) are equal. In particular a Gröbner basis of the
ideal �χi (M) of K[Ti] with respect to term order ≺ on Ti is also a Gröbner basis
of the ideal �χ(M) of A with respect to the term order ≺ on T.
Proof. Note first that if we see �χ(M) as a K-vector space it is clear that
�χ(M) = �χi (M) ⊕ K[Td]. Pick a non-null polynomial f ∈ �χ(M) and let
eX1 := lp≺(f). So eX1 ∈ �i(M) if X1 ∈ IND(M), or eX1 ∈ K[Td] \ 0 if X1 is
a dependent set of M. If X1 ∈ IND(M) then eX1 ∈ Lt≺(�χ(M)). Suppose now that
X1 is a dependent set of M. Then there is a circuit C ⊆ X1. From Definition 2.2 we
know that ∂eC ∈�χ(M). It is clear that eC ∈ Lt≺(pi(�χ(M))) and so we have also
eX1 ∈ Lt≺(pi(�χ(M))).
Remark 3.6 It is well known that a term order ≺ of T(A) determines also a unique
reduced Gröbner basis of �χ(M) denoted (Gr )≺. From the definitions we can deduce
also that, for every pair of term orders ≺ and ≺′ on T(A),

BG≺ = BG≺′ ⇔ (Gr)≺ = (Gr )≺′ ⇔ Lt≺
(
�χ(M)

)
= Lt≺′

(
�χ(M)

)
.

Definition 3.7 For a term order ≺ on T(A) we say that π≺ ∈ Sn, is the permutation
compatible with ≺ if, for every pair i,j ∈ [n], we have

ei ≺ ej iff i <π≺ j
(
⇔ π≺−1(i) < π≺−1(j)

)
.

Let Cπ≺ be the subset of circuits of M such that:
◦ C ∈ Cπ≺ iff inf<π≺ (C) = απ(C)

(
= inf<π≺ (cl(C)\C)

)
and C\απ(C) is inclusion

minimal with this property.

In the following we replace “π≺” by “π” if no mistake can results. We recall that
given a unidependent set U of the matroid M([n]), C(U) denotes the unique circuit
of M contained in U.
Theorem 3.8 Let ≺ be a term order on T(A) compatible with the permutation π ∈
Sn. Then the family Gr :=

{
∂eC(U) : U ∈ Cπ≺ (M)

}
form a reduced Gröbner basis

of �χi (M) with respect to the term order ≺.
Proof. From Proposition 3.5 it is enough to prove that (Gr)≺ is a reduced Gröbner
of �χi (M). Let f be any element of �χi (M), we have from Theorem 2.3 that

f =
∑

U∈Uπ
ξU∂eU, ξU ∈ K�.

Let now remark that lp≺
(
∂eU

)
= eU\απ (U) and that these terms are all different. We

have then clearly that

lp≺(f) = sup≺
{
lp≺(∂eU ) : U ∈ Uπ

}
.



10 Raul Cordovil and David Forge
7, 2(2005)

Given an arbitrary U′ ∈ Uπ(M) it is clear that απ(C(U′)) = απ(U′). So,

C(U′) \απ(C(U′)) ⊆ U′ \απ(U′).

Let C′ be a circuit of Cπ such that C′ \απ(C′) ⊆ C(U) \απ(C(U)). So we have that
lp≺(∂eC′ ) divides lp≺(∂eU ), and (Gr )≺ is a Gröbner basis.

Suppose for a contradiction that (Gr)≺ is not a reduced Gröbner basis: i.e., there
exists two circuits C and C′ in Cπ and an element c ∈ C such that eC′\απ (C′) divides
eC\c

(
⇔ C′ \ απ(C′) ⊆ C \ c

)
. First we can say that c �= απ(C) because the

sets C′ \ απ (C′) and C \ απ(C) are incomparable. This in particular implies that
απ(C) ∈ C′ \ απ (C′), and απ(C′) ≺ απ(C). On the other hand we have απ(C′) ∈
cl
(
C′ \απ(C′)

)
⊆ cl(C \ c) = cl(C \απ(C)), so απ(C) ≺ απ(C′), a contradiction.

Corollary 3.9 The set Gu :=
{
∂eC : C ∈ C(M)} is a minimal universal Gröbner

basis of the ideal �χ(M).
Proof. From Theorem 3.8, the reduced Gröbner bases constructed for the different
orders ≺ are all contained in Gu. We prove the minimality by contradiction. Let
C0 = {i1, . . . , im} be a circuit of M and let π ∈ Sn be a permutation such that
π−1(ij ) = j, j = 1, . . . ,m. Then G′u := {∂eC : C ∈ C \C0} it is not a Gröbner basis
because lp≺π (∂eC0 ) = eC0\i1 is not in Lt≺π (G′u).
To finish this section we give an important characterization of the no broken circuit
bases of the χ-algebras in terms of the Gröbner bases of their ideals.

Definition 3.10 Consider a permutation π ∈ Sn and the associated re-ordering <π
of [n]. When the <π-smallest element inf<π (C) of a circuit C ∈ C(M), |C| > 1, is
deleted, the remaining set, C \ inf<π (C), is called a π-broken circuit of M. We say
that

π-nbc(M) := {eX : X ⊆ [n] contains no π-broken circuit of M}
is the π-no broken circuit bases of Aχ(M). As the algebra Aχ(M) does not depend
of the ordering of the elements of M it is clear that π-nbc(M) is a no broken circuit
bases of Aχ(M).

Corollary 3.11 Let B be a basis of the module Aχ(M). Then are equivalent:
(3.11.1) B is the canonical basis B≺, for some term order ≺ on T(A).
(3.11.2) B is the π-no broken circuit bases π-nbc(M), for some permutation π ∈

Sn.

(3.11.3) B is the canonical basis BGr, for some reduced Gröbner basis Gr of the
ideal �χ(M).

Proof. (3.11.1) ⇒ (3.11.2) Let ≺ be a term order of T(A). Since Gu is a universal
Gröbner basis of �χ(M) (see Corollary 3.9) it is trivially a Gröbner basis relatively
to ≺. We have already remarked that the leading term of ∂eC is eC\c where c =
inf<π≺ (C). From Proposition 3.4 we conclude that B≺ = π≺-nbc(M).



7, 2(2005)
Gröbner and diagonal bases in Orlik-Solomon type algebras 11

(3.11.2) ⇒ (3.11.3) Suppose that B = π-nbc(M). Let ≺π be the degree lexicographic
order of T determined by the permutation π ∈ Sn. Note that π≺π = π. ¿From
Theorem 3.8 we know that (Gr )≺π =

{
∂eC : C ∈ C≺π} is the reduced Gröbner basis

of �χ(M) with respect to the term order ≺π. Then B is the canonical basis for the
reduced Gröbner basis (Gr )≺π .
(3.11.3) ⇒ (3.11.1) It is a consequence of Proposition 3.4 and Remark 3.6.

4 Diagonal bases of χ-algebras

Proposition 4.1 Let Aχ(M) be a χ-algebra with the associated map χ : 2[n] → K.
For any non loop element x of M([n]), we define the two maps:

χM\x : 2
[n]\x → K by χM\x(X) = χ(X) and (4.1)

χM/x : 2
[n]\x → K by χM/x(X) = χ(X ◦x). (4.2)

There are two χ-algebras, AχM/x (M/x) and AχM\x (M\x), associated to the maps
χM\x and χM/x, respectively.

Proof. From (2.2.1) we know that χ(X) �= ∅ iff X ∈ IND(M). The deletion case
being trivial, we will just prove the contraction case. We have to show that χM/x
verifies properties (2.2.1) and (2.2.2). The first property is verified since a set I is
independent in M/x iff I ∪x is independent in M. To see that the second property
is also verified, let U and U′ be two unidependents sets of M/x. I.e., iff U ∪ x and
U′ ∪x are two unidependents sets of M. From (2.2.1) we know that

∂eU∪x = εU∪x,U′∪x (∂eU′∪x)eU\U′ where εU∪x,U′∪x ∈ K∗.
Let ∂′ be the χM/x-boundary, i.e., the linear mapping ∂

′ : A/〈ex〉 → A/〈ex〉 such
that for ever ei ∈ E\x we have ∂′ei = 1, ∂′e∅ = 1 and for every monomial eX, x �∈ X
and X �= ∅,

∂′eX =
p=m∑
p=1

(−1)pχM/x(X \ ip)eX\ip =
p=m∑
p=1

(−1)pχ(X \ ip ◦x)eX\ip.

To finish the proof we will show that there is a scalar ε̃
U,U′ ∈ K∗ such that

∂′eU = ε̃U,U′ (∂
′eU′ )eU\U′.

Let X,X′ ⊆ [n] be two disjoint subsets. From Definition 2.1 we known that

eX ·eX′ = βX,X′eX∪X′, where βX,X′ =
∏

βi,j, (ei ∈ X,ej ∈ X′ and i > j).

So we have with U = (i1, . . . , im) and U′ = (j1, . . . ,jk), U ∩U′ = ∅, x �∈ U ∪U′:

±∂eU∪x =
p=m∑
p=1

(−1)pχ(U \ ip ◦x)eU∪x\ip + (−1)m+1χ(U)eU,



12 Raul Cordovil and David Forge
7, 2(2005)

∂′eU =
p=m∑
p=1

(−1)pχ(U \ ip ◦x)eU\ip ,

±(∂eU′∪x)eU\U′ =
p=k∑
p=1

(−1)pχ(U′ \ jp ◦x) ·β ·eU∪x\jp + (−1)k+1χ(U′) ·β′ ·eU,

where β = β
U′∪x\jp ,U\U′

and β′ = β
U′,U\U′ .

(∂′eU′ )eU\U′ =
p=k∑
p=1

(−1)pχ(U′ \ jp ◦x) ·βU′\jp ,U\U′ ·eU\jp.

After remarking that
β

U′∪x\jp ,U\U′
β−1

U′\jp ,U\U′
= β

x,U\U′

does not depend on jp, we can deduce that

∂′eU = ε̃U,U′ (∂
′eU′ )eU\U′ with ε̃U,U′ = ±εU∪x,U′∪x ·β−1x,U\U′.

Proposition 4.2 For every non loop element x of M([n]), there is a unique monomor-
phism of vector spaces, ix : A(M \ x) → A(M), such that such that for every
I ∈ IND(M\x), we have ix(eI ) = eI .
Proof. By a reordering of the elements of the matroid M we can suppose that x = n.
It is clear that

NBC(M\x) =
{
X : X ⊆ [n− 1] and X ∈ NBC(M)

}
,

so the proposition is a consequence of Equation (4.1).

Proposition 4.3 For every non loop element x of M([n]), there is a unique epimor-
phism of vector spaces, px : A(M) → A(M/x), such that, for every eI, I ∈ IND(M),
we have

px(eI ) :=

⎧⎪⎪⎨
⎪⎪⎩
eI\x if x ∈ I,
χ(I\y,x)
χ(I\y,y) eI\y if there is y ∈ I parallel to x,
0 otherwise.

(4.3)

Proof. From Remark 2.3, it is enough to prove that px(∂eU ) = 0, for all unidependent
U = (i1, . . . , im). We recall that if x ∈ U then U \x is a unidependent set of M/x.
There are only the following four cases:

◦ If U contains x but no y parallel to x then:
±px(∂eU ) = px((−1)mχ(U \x)eU\x +

∑
ip∈U\x

(−1)pχ(U \{ip,x}◦x)eU\ip ))

=
∑

ip∈U\x
(−1)pχ(U \{ip,x}◦x)eU\{ip ,x} = 0

from Proposition 4.1.



7, 2(2005)
Gröbner and diagonal bases in Orlik-Solomon type algebras 13

◦ If U does not contain x but contains a y parallel to x then:

±px(∂eU ) = px
(
(−1)mχ(U \y)eU\y +

∑
ip∈U\y

(−1)pχ(U \{ip,y}◦y)eU\ip
)

=
∑

ip∈U\y
(−1)pχ(U \{ip,y}◦y)

χ(U \{ip,x}◦x)
χ(U \{ip,y}◦y)

eU\{ip,y} = 0

like previously since U \y is again a unidependent of M/x.
◦ If U contains x and a y parallel to x then:

±px(∂eU ) = px(χ(U \{x,y}◦y)eU\x −χ(U \{x,y}◦x)eU\y )

= χ(U \{x,y}◦y)χ(U \{x,y}◦x)
χ(U \{x,y}◦y)eU\{x,y} −χ(U \{x,y}◦x)eU\{x,y} = 0.

◦ If U does not contain x nor a y parallel to x then:

px(∂eU ) = px
( ∑

ip∈U
(−1)pχ(U \ ip)eU\ip

)
= 0.

Theorem 4.4 For every element x of a simple M([n]), there is a splitting short exact
sequence of vector spaces

0 → A(M\x) ix−→ A(M) px−→ A(M/x) → 0. (4.4)

Proof. From the definitions we know that the composite map px◦ ix, is the null map
so Im(ix) ⊆ Ker(px). We will prove the equality dim(Ker(pn)) = dim(Im(in)). By a
reordering of the elements of [n] we can suppose that x = n. The minimal broken
circuits of M/n are the minimal sets X such that either X or X ∪{n} is a broken
circuit of M (see the Proposition 3.2.e of [5]). Then

NBC(M/n) =
{
X : X ⊆ [n− 1] and X ∪{n}∈ NBC(M)

}
and

NBC(M) = NBC(M\n)
⊎{

I ∪n : I ∈ NBC(M/n)
}
. (4.5)

So dim(Ker(pn)) = dim(Im(in)). There is a morphism of modules

p−1n : A(M/n) → A, where p−1n ([I]A(M/n)) := [I ∪ n]A, ∀I ∈ NBC(M/n).

It is clear that the composite map pn◦ p−1n is the identity map. From Equation (4.5)
we conclude that the exact sequence (4.4) splits.
Similarly to [17] (see also [4]), we now construct, making use of iterated contrac-
tions, the dual bases nbc∗� = (b

∗
i ) of the bases nbc � := (bj ) of the vector space A�.

More precisely nbc∗� is the basis of A
∗
� the vector space of the linear forms such that

〈b∗i ,bj〉 = δij (the Kronecker delta).



14 Raul Cordovil and David Forge
7, 2(2005)

We associate to the ordered independent set Iσ := (iσ(1), . . . , iσ(p)) of M the
linear form on A�, pIσ : A� → K, defined as the composite of the maps peiσ(p) ,
peiσ(p−1)

, . . . , peiσ(1)
, i.e.,

pIσ := peiσ(1)
◦ peiσ(2)

◦ · · · ◦ peiσ(p) . (4.6)

We call pIσ the iterated residue with respect to the ordered independent set I
σ. We

remark that the map pIσ depends on the order chosen on I
σ and not only on the

underlying set I. We associate to Iσ the flag of flats of M,

Flag(Iσ) := c�
(
{iσ(p)}

)
� c�

(
{iσ(p), iσ(p−1)}

)
� · · · � c�

(
{iσ(p), . . . , iσ(1)}

)
.

Proposition 4.5 Let J ∈ IND�(M) then we have pIσ (eJ ) �= 0 iff there is a unique
permutation τ ∈ S� such that Flag (Jτ ) = Flag (Iσ). And in this case we have
pIσ (eJ ) = χ(I

σ)/χ(Jτ ). In particular we have pIσ (eI ) = 1 for any independent set
I and any permutation σ.

Proof. The first equivalence is easy to prove in both direction. To obtain the expres-
sion of pIσ (eJ ) we just need to iterate � times the residue. This gives:

pIσ (eJ ) =
χ(J \ jτ (�) ◦ iσ(�))
χ(J \ jτ (�) ◦ jτ (�))

× χ(J \{jτ (�),jτ (�−1)}◦ iσ(�−1) ◦ iσ(�))
χ(J \{jτ (�),jτ (�−1)}◦ jτ (�−1) ◦ iσ(�))

×···

· · ·× χ(I
σ)

χ(jτ (1) ◦ Iσ \ iσ(1))
.

After simplification we obtain the announced formula. The last result is clear.

Remark 4.6 The fact that pIσ (eJ ) is null depends on the permutation σ. For ex-
ample, for any simple matroid of rank 2 we have p13(e12) = 0 and p31(e12) �= 0.
But if pIσ (eJ ) �= 0 then its value does not depend on σ. We mean by this that if
there are two permutations σ and σ′ such that pIσ (eJ ) �= 0 and pIσ′ (eJ ) �= 0 then
pIσ (eJ ) = pIσ′ (eJ ).

Definition 4.7 ([17]) We say that the subset I� ⊆
{
[I]A : I ∈ IND�(M)} is a

diagonal basis of A� if and only if the following three conditions hold:

(4.7.1) For every [I]A ∈ I� there is a fixed permutation of the set I denoted σI ∈ S�;

(4.7.2)
∣∣I�| ≥ dim(A�);

(4.7.3) For every [I]A, [J]A ∈ I� and every permutation τ ∈ S�, the equality Flag (Jτ ) =
Flag (IσI ) implies J = I.

Theorem 4.8 Suppose that I� is a diagonal basis of A�. Then I� is a basis of A� and
I∗� := {pIσI : [I]A ∈ I�} is the dual basis of I�.



7, 2(2005)
Gröbner and diagonal bases in Orlik-Solomon type algebras 15

Proof. Pick two elements [I]A, [J]A ∈ I�. Note that pIσI (eJ ) = δIJ (the
Kronecker delta), from Condition (4.7.2) and Proposition 4.5. The elements of
I� are linearly independent: suppose that [J] =

∑
ζj [Ij ], ζj ∈ K \ 0; then 1 =

pJ σJ ([J]) = pJ σJ
(∑

ζj [Ij ]
)

= 0, a contradiction. It is clear also that I∗� is the
dual basis of I�.

The following result gives an interesting explanation of results of [6, 7].

Corollary 4.9 nbc�(M) is a diagonal basis of A� where σI is the identity for every
[I]A ∈ nbc�(M). For a given [J]A ∈ A�, suppose that

(4.9.2) [J]A =
∑
ξ(I,J)[I]A, where [I]A ∈ nbc�(M) and ξ(I,J) ∈ K.

Then are equivalent:

◦ ξ(I,J) �= 0,
◦ Flag (I) = Flag (Jτ ) for some permutation τ.

If ξ(I,J) �= 0 we have ξ(I,J) = χ(I)
χ(J τ )

. In particular if A is the Orlik-Solomon algebra
then ξ(I,J) = sgn (τ).

Proof. By hypothesis (4.7.1) and (4.7.2) are true. We claim that nbc �(M) verifies
(4.7.3). Suppose for a contradiction that J �= I, [J]A, [I]A ∈ nbc �(M) and there is
τ ∈ S�, such that Flag (Jτ ) = Flag (I). Set I = (i1, . . . , i�) and J = (jτ (1), . . . ,jτ (�)),
and suppose that jτ (m+1) = im+1, . . . ,jτ (�) = i� and im �= jτ (m). Then there is a
circuit C of M such that

im,jτ (m) ∈ C ⊆{im,jτ (m), im+1, im+2, . . . , i�}.
If jτ (m) < im [resp. im < jτ (m)] we conclude that

I �∈ NBC�(M) [resp. J �∈ NBC�(M)]
a contradiction. So nbc �(M) is a diagonal basis of A�.

From Theorem 4.8 we conclude that nbc∗� :=
{
pI : [I]A ∈ nbc} is the dual basis

of nbc. Suppose now that [J]A =
∑
ξI [I]A, where [I]A ∈ nbc�(M) and ξI ∈ k. Then

ξI = pI (eJ ) and the remaining follows from Proposition 4.5.
Making full use of the matroidal notion of iterated residue, see Equation (4.6), we

are able to prove the following result very close to Proposition 2.1 of [18].

Proposition 4.10 Consider the set of vectors V := {v1, . . . ,vk} in the plane xd = 1
of Kd. Set AK := {Hi : Hi = Ker(vi) ⊆ (Kd)∗, i = 1, . . . ,k} and let OT(AK) be
its Orlik-Solomon-Terao corresponding algebra. Fix a diagonal basis I� ⊆ {[I]A : I ∈
IND�(M)} of A� and let I∗� = {pIσI : [I]A ∈ I�} be the corresponding dual basis.
Then, for any eJ ∈ A� \ 0, we have

∑
I∈I�

pIσI (eJ ) =
∑
I∈I�

〈
pIσI ,eJ

〉
= 1.



16 Raul Cordovil and David Forge
7, 2(2005)

Proof. We have for any � + 1-subset of V, ∑p=�+1p=1 (−1)pχ(U \ ip) = 0. (This is
the development of a determinant with two lines of 1.) For any rank � unidependent
U = {i1, . . . , i�+1} of the matroid M(AK), we have

∂eU =
p=�+1∑

p=1

(−1)pχ(U \ ip)eU\ip.

Since the sum of the coefficients in these relations is 0 and that these relations are
generating, see Remark 2.3, we can deduce that the sum of the coefficients in any
relation in OT(AK) is also equal to 0 which concludes the proof.

5 Examples

In this section we will show on a small example the different results of the three
previous sections.

Consider the the set of 6 points {p1, . . . ,p6} in the affine plane z = 1 of three
dimensional real vector space R3, whose coordinates are indicated in Figure 1. Set
vi :=

−−−→
(0,pi), i = 1, . . . , 6. And let A be the corresponding hyperplane arrangement

of (R3)∗, A := {Hi = Ker(vi), i = 1, . . . , 6}. Let M(A) [resp. M(A)] be the corre-
sponding rank three [resp. oriented] matroid. So like in Example 2.4, the arrangement
A defines the three classical Orlik-Solomon type algebras: the original Orlik-Solomon
algebra OS(M(A)) through M(A), the Orlik-Solomon-Terao algebra OT(A) directly
from the vi and the Cordovil algebra A(M(A)) from M(A).

p1

p3

p2

p4 p5

p6

� �

�

�

�

�(0,0,1)

(0, 1
2
, 1)

(0,1,1)

( 1
2
, 0, 1) (1,0,1)

( 1
3
, 1

3
, 1)

����������

�
�
�
�
�
�
�
�
�
�

Figure 1: The rank 3 matroid on the set {p1, . . . ,p6}.
Let Aχ be a χ-algebra on M(A). We know that

nbc 3 = {e124,e125,e126,e134,e135,e136}

together with σ124 = σ125 = σ134 = σ135 = σ136 = σ156 = id is a diagonal ba-
sis of A3, from Corollary 4.9. Directly from the Definition 4.7 we see that B3 =
{e124,e125,e134,e135,e136,e156} with σ124 = σ134 = σ135 = σ136 = σ156 = id and



7, 2(2005)
Gröbner and diagonal bases in Orlik-Solomon type algebras 17

σ125 = (132) is also a diagonal basis of A3. We will look at expressions on the ba-
sis nbc3 (resp. B3) of the vector space A3, of some elements of the type eB, B
basis of M(A), for the three χ-algebras of Example 2.4. Especially, we will ver-
ify as stated in Remark 4.6 that p125id (e235) = p125(132) (e235). Let also point out
that for the Orlik-Solomon-Terao algebra, we will have have

∑
I∈B pIσ (eJ ) = 1 as

proved in Proposition 4.10. Finally recall that T is set of the monomials of A and set
T� := {eX ∈ T : |X| = �}.
(a) Let us first take the Orlik-Solomon algebra OS(M(A)) :

From Remark 2.3 , the basis of OS(M(A)) is simply the nbc-bases:
nbc(M) = T0 ∪ T1 ∪ nbc2 ∪ nbc3,

with nbc2 = {e12,e13,e14,e15,e16,e24,e25,e26,e34,e35,e36}, and
nbc3 = {e124,e125,e126,e134,e135,e136}.

The basis of �χ(M(A)) is the union of the dependents and of the boundaries of
the inactive unidependents:

∂uni3 ∪ dep3 ∪∂uni4 ∪ T4 ∪ T5 ∪ T6
where ∂uni3 = {∂e123,∂e145,∂e256,∂e346}, dep3 = {e123,e145,e256,e346} and
∂uni4 is the set

{∂e1234,∂e1235,∂e1236,∂e1245,∂e1246,∂e1256,∂e1345,∂e1346,∂e1356,∂e1456}.
Note that we have

|nbc2| + |∂uni3| = 11 + 4 = 15 = dim(A2)
and

|nbc3| + |∂uni4| + |dep3| = 6 + 10 + 4 = 20 = dim(A3).
Take first on [n] the natural order. We have then for the leading term ideal

Lt<(G) = 〈eBC : BC broken circuit〉.
We obtain explicitly:

Lt<(G) = 〈e23,e45,e56,e46,e246,e345,e356〉.
Always for the natural order, from Theorem 3.8, we obtain for the reduced
Gröbner basis:

Gr =
{
∂e123,∂e145,∂e256,∂e346

}
.

If we take now the term order ≺π on T(A), defined by the permutation π :=
(234561), we get now:

Lt≺(G) = 〈e13,e15,e56,e46,e146,e345,e165〉,



18 Raul Cordovil and David Forge
7, 2(2005)

and then for the corresponding reduced Gröbner basis:

Gr =
{
∂e123,∂e145,∂e256,∂e346,∂e2345

}
.

Finally from Corollary 3.9, we get the minimal universal Gröbner basis

Gu =
{
∂eC : C ∈ C(M)}.

We obtain explicitly:

Gu = {∂e123,∂e145,∂e256,∂e346,∂e1246,∂e1356,∂e2345}.

Now we will use the results of Section 4 to express pure elements in different diag-
onal bases. Consider the diagonal basis nbc3 of the K-vector space OS(M(A))3.
So we have:

e156 = sgn(165)e125 + sgn(156)e126 = −e125 + e126
and

e235 = sgn (325)e125 + sgn (235)e135 = −e125 + e135.
For the diagonal basis B3 of the K-vector space OS(M(A))3, we have:

e126 = sgn(162)sgn(152)e125 + sgn(126)e156 = e125 + e156

and
e235 = sgn(152)sgn(352)e125 + sgn(235)e135 = −e125 + e135.

(b) Let us take the Orlik-Solomon-Terao algebra OT(A) :
For the different bases and Gröbner bases we obtain formally the same results.
There is in fact differences which are hidden by the operator ∂ (indeed ∂ is
function of χ).

For the diagonal basis nbc3 of the K-vector space OT(A)3 we have:

e156 =
det(125)
det(165)

e125 +
det(126)
det(156)

e126 =
3
2
e125 −

1
2
e126

and
e235 = sgn (325)e125 + sgn (235)e135 = −e125 + e135

For the diagonal basis B3 of the K-vector space OT(A)3 we have:

e126 =
det(152)
det(162)

e125 +
det(156)
det(126)

e156 = 3e125 − 2e156.

and

e235 =
det(152)
det(352)

e125 +
det(135)
det(235)

e135 = −e125 + 2e135.



7, 2(2005)
Gröbner and diagonal bases in Orlik-Solomon type algebras 19

(c) Let us take the Cordovil Z-algebra A(M(A)) :
For the diagonal basis nbc3 of the K-vector space A(M(A))3 we have:

e156 = χ(125)χ(165)e125 + χ(126)χ(156)e126 = e125 −e126
and

e235 = sgn (325)e125 + sgn (235)e135 = −e125 + e135.
For the diagonal basis B3 of the K-vector space A(M(A))3 we have:

e126 = χ(152)χ(162)e125 + χ(156)χ(126)e156 = e125 −e156
and

e235 = sgn(152)sgn(352)e125 + sgn(235)e135 = −e125 + e135.

Received: September 2003. Revised: January 2004.

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