December 2, 2003 17:17 WSPC/132-IJAC 00160

International Journal of Algebra and Computation

Vol. 13, No. 5 (2003) 517–526
c World Scientific Publishing Company

GRADED IDENTITIES FOR THE ALGEBRA OF n × n

UPPER TRIANGULAR MATRICES OVER AN INFINITE FIELD

PLAMEN KOSHLUKOV∗
IMECC, UNICAMP, Cx. P. 6065, 13083-970 Campinas, SP, Brazil
plamen@ime.unicamp.br

ANGELA VALENTI†
Dipartimento di Matematica e Applicazioni
Università di Palermo, Via Archirafi, 34, 90123 Palermo, Italy
avalenti@math.unipa.it

Received January 2002

Communicated by A. Ol’shanskii

We consider the algebra Un (K) of n × n upper triangular matrices over an infinite field
K equipped with its usual Zn -grading. We describe a basis of the ideal of the graded
polynomial identities for this algebra.

Keywords: Graded identities; basis of identities; graded codimensions.

AMS Mathematics Subject Classification (2000): 16P90, 16R10, 16R20, 16R50, 16W50

0. Introduction
The interest into graded polynomial identities was inspired by their importance for
the structure theory of PI algebras (see for example [5] or [16]). Shortly afterwards
they became an object of independent studies. This in turn was determined by the
various applications that the graded identities can find in PI theory. We mention
some of the most important results concerning graded polynomial identities. Thus,
for example, in [3] and in [10], it was proved that if G is a finite group and A is a
G-graded algebra, then A is PI if and only if its 0-component is PI. In [3, 6–9, 12,
13], several numerical characteristics of T-ideals were transferred to the graded case
and lots of their properties were deduced. In [11], bases of the 2-graded identities
for several important algebras were described. As a corollary it was obtained a
rather elementary proof of the fact that the algebras M11 (G) and G ⊗ G satisfy the

∗ Partially supported by CNPq and by FAEP, UNICAMP.

† Partially supported by MURST of Italy.

517
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518 P. Koshlukov & A. Valenti

same polynomial identities when the base field is of characteristic 0. Here G stands
for the infinite dimensional Grassmann algebra, and M11 (G) is the algebra of the
2 × 2 matrices over G whose main diagonal elements are even (central) elements
of G, and whose other diagonal elements are odd elements of G. In [19], a basis of
the n-graded identities of the algebra of n × n matrices Mn (K) was found when
K is a field of characteristic 0, and in [20], this was extended to the Z-gradings of
Mn (K). Furthermore, in [17], the results of [11] were extended to algebras over an
infinite field of characteristic p 6= 2 (and as a consequence it was obtained a really
elementary proof of the coincidence of the T-ideals of M11 (G) and G ⊗ G). In [2], it
was established that the main result of [19] holds for algebras over an infinite field.
It is worth mentioning that the study of graded identities was one of the key
ingredients in Kemer’s methods for developing the structure theory of PI algebras
and in particular, for resolving positively the Specht problem in characteristic 0.
Graded identities, along with other kinds of “weaker” identities play an essential role
in study of the polynomial identities satisfied by concrete algebras. Some applica-
tions of graded identities satisfied by concrete algebras can be found in [11, 13, 17].
The polynomial identities satisfied by the algebra Un (K) of the n × n upper
triangular matrices are of particular interest. It is well known that for every field
K and every n they are finitely based (as T-ideal) (see for example [21]). Thus,
when the field K is infinite, the T-ideal of Un (K) is generated by the identity
[x1 , x2 ][x3 , x4 ] · · · [x2n−1 , x2n ] where [a, b] = ab − ba is the usual commutator. It is
also well known that the identities of Un (K) are closely related to the problem of
the description of the subvarieties of the variety of associative algebras generated
by M2 (K). Since the latter is still open when charK 6= 0, it is important to obtain
further information about the identities in Un (K). A detailed description of the
2-graded identities satisfied by the algebra U2 (K) when charK = 0 is given in [18].
In this paper we describe a basis of the n-graded polynomial identities for the
algebra Un (K) over an infinite field K and as an application we compute the
asymptotics for the sequence of its graded codimensions.
These results generalize the ones in [18]. We hope they can be useful in describing
the behaviour of the subvarieties of the variety generated by the matrix algebra of
order 2 over an infinite field of characteristic, not 2.

1. Preliminaries
We fix an infinite field K, and consider all algebras (graded and ungraded), vector
spaces etc. over K. Denote by Un (K) the algebra of n×n upper triangular matrices
over K i.e., n × n matrices with zero entries below the main diagonal. The algebra
Un (K) has a natural Zn -grading Un (K) = V0 ⊕ V1 ⊕ · · · ⊕ Vn−1 with

Vi = {a1,i+1 e1,i+1 + a2,i+2 e2,i+2 + · · · + an−i,n en−i,n |ar,s ∈ K} , 0≤i≤n−1

where the eij are the elementary matrices having 1 as (i, j)th entry and 0 elsewhere.
In particular V0 consists of all diagonal matrices. Clearly all Vi are subspaces of
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Graded Identities for the Algebra of n × n Upper Triangular Matrices 519

Un (K) and Vi Vj ⊆ Vi+j where i + j is taken modulo n. Observe that if i + j ≥ n

then Vi Vj = 0.
Let X be a countable infinite set, and let K(X) be the free associative algebra
freely generated by X over K. Suppose that X = X0 ∪ X1 ∪ · · · ∪ Xn−1 where
Xi ∩ Xj = ∅ if i 6= j, all Xi being infinite, and Xi = {xi1 , xi2 , . . .}. The weight of
the variable xij is w(xij ) = i, and if m = xi1 j1 xi2 j2 · · · xik jk is a monomial then
its weight is defined as w(m) = i1 + i2 + · · · + ik , the last sum is taken modulo
n. Define K(X)i as the span of all monomials of weight i, 0 ≤ i ≤ n − 1, then
K(X) = K(X)0 ⊕ K(X)1 ⊕ · · · ⊕ K(X)n−1 is a Zn -graded algebra. We shall use
the term n-graded instead of Zn -graded. This algebra is the free n-graded algebra
in the sense that given an n-graded algebra A = A0 ⊕ A1 ⊕ · · · ⊕ An−1 , every
map ϕ: X → A such that ϕ(Xi ) ⊆ Ai can be extended uniquely to a graded
homomorphism Φ: K(X) → A.
Let f = f (xi1 j1 , xi2 j2 , . . . , xim jm ) ∈ K(X) be a polynomial, and let A be an
n-graded algebra. Then f is an n-graded identity for the algebra A if f (ai1 j1 ,
ai2 j2 , . . . , aim jm ) = 0 for every aik jk ∈ Aik . In other words f becomes zero when we
substitute its variables for homogeneous elements of A having the same weight as
the respective variables. Obviously the set Idgr (A) of all n-graded identities is an
ideal of K(X), and this ideal is stable under all graded endomorphisms of A. We
call such ideals Tn -ideals. The class of all n-graded algebras satisfying the graded
identities of Idgr (A) is the variety var(A) of n-graded algebras determined by A
(and by Idgr (A)).
The quotient algebra K(X)/Idgr (A) is called the relatively free n-graded alge-
bra. We shall use the same symbols xij instead of xij + Idgr (A) for the generators
of the relatively free graded algebra. Then if B ∈ var(A), every homogeneous map
ϕ: X → B (i.e. a map that preserves the grading) can be extended uniquely to a
homomorphism of graded algebras Φ: K(X)/Idgr (A) → B.
Since our field is infinite then every ideal of n-graded identities is generated
by its multihomogeneous elements. Note that when the base field is of positive
characteristic it may occur that the multilinear identities (graded identities) of an
algebra do not generate the T-ideal (the Tn -ideal, respectively) of the given algebra.

2. The Graded Identities of Un (K)

Throughout this section A = Un (K) is the algebra of n×n upper triangular matrices
with its natural Zn -grading Un (K) = V0 ⊕ V1 ⊕ · · · ⊕ Vn−1 .
Now if S ⊆ K(X) is any set of polynomials, we denote by hSiTn the Tn -ideal of
K(X) generated by S. We start with the following lemma.
Lemma 2.1. The algebra Un (K) satisfies the graded identities
x01 x02 − x02 x01 ≡ 0 (1)
xi 1 1 xi 2 2 ≡ 0 (2)
whenever i1 + i2 ≥ n.
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520 P. Koshlukov & A. Valenti

Proof. The identity x01 x02 − x02 x01 ≡ 0 holds in Un (K) since two diagonal ma-
trices commute. Moreover since i + j ≥ n implies Vi Vj = 0, the graded identities
xi1 1 xi2 2 ≡ 0 with i1 + i2 ≥ n, hold for Un (K) as well.

Denote by I the ideal of n-graded identities generated by the identities (1) and
(2), and set J = Idgr (Un (K).
We wish to show that I = J. We start with an obvious observation.
Remark 2.1. For any i1 +i2 +· · ·+ik ≥ n, xi1 1 xi2 2 · · · xik k ≡ 0 is a graded identity
of Un (K).

Proof. It follows from (2) by the invariance of Tn -ideals under endomorphisms

preserving the grading.

Consider the set of monomials of K(X) of the type

u = w0 xk1 i1 w1 · · · wt−1 xkt it wt (3)
where k1 + · · · + kt < n and w0 , . . . , wt are (possibly empty) monomials in the
variables x0i of homogeneous degree 0 and in each wi these variables are written in
increasing order from left to right.
The next theorem gives the multilinear structure of the relatively free n-graded
algebra in the variety determined by Un (K) and as a consequence, a basis of the
n-graded identities of Un (K).
Theorem 2.1. If K is an infinite field, the monomials (3) are a basis of K(X)
modulo Idgr (Un (K). Moreover Idgr (Un (K)) = h[x01 , x02 ], xi1 xj2 |i + j ≥ niTn .

Proof. We first claim that the monomials (3) are linearly independent modulo
Idgr (Un (K)).
αi ui ∈ Idgr (Un (K)) for some αi ∈ K where the
P
In fact suppose that f =
monomials ui are of the type (3). Since K is an infinite field, graded ideals are
multihomogeneous. Hence we may assume that all ui are multihomogeneous of
the same degree sequence, i.e. the same variables appear in each monomial ui .
Fix one monomial with nonzero coefficient say u1 and let u1 = w0 xk1 i1 w1 · · · wt−1
xkt it wt . We now evaluate f on Un (K) as follows: evaluate each variable appear-
ing in w0 , into e11 , the variable xk1 i1 into e1,k1 +1 , each variable appearing in w1
into ek1 +1,k1 +1 , the variable xk2 i2 into ek1 +1,k1 +k2 +1 , . . . , the variable xkt it into
ek1 +···+km−1 +1,k1 +···+km +1 and each variable in wt into ejj where j = k1 + · · · +
km + 1. It follows that u1 evaluates into e1j and all other monomials of f evaluate
to zero. Thus f 6≡ 0 on Un (K), a contradiction.
Next we claim that, modulo h[x01 , x02 ], xi1 xj2 |i + j ≥ niTn , every element of
K(X) can be written as a linear combination of monomials of type (3). In fact, this
is clear since Tn -ideals are invariant under homogeneous substitutions.
It follows that the monomials (3) are a basis of K(X) modulo Idgr (Un (K) and
that Idgr (Un (K)) = h[x01 , x02 ], xi1 xj2 , i + j ≥ niTn .
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Graded Identities for the Algebra of n × n Upper Triangular Matrices 521

3. Generic Matrices and Graded Identities

We now introduce a model for K(X)/Idgr (Un ), the relatively free n-graded algebra.
This model is based on a modification of the construction of the ring of generic
matrices.
Let yijk , i, j, k ≥ 0, be commuting variables and consider the polynomial algebra
K[yijk ] in these variables. The tensor product Un (K) ⊗K K[yijk ] is canonically
isomorphic to the algebra Un (K[yijk ]). Let

Yik = y1,i+1,k e1,i+1 + y2,i+2,k e2,i+2 + · · · + yn−i,n,k en−i,n

with 0 ≤ i ≤ n−1, k = 1, 2, . . . , and set Gn the subalgebra of Un (K[yijk ]) generated

by the matrices Yik , 0 ≤ i ≤ n − 1, k = 1, 2, . . . The algebra Gn is n-graded in
a natural way, precisely the same as Un (K). Furthermore it satisfies all graded
identities of Un (K).

Lemma 3.1. The algebra Gn is isomorphic as a graded algebra to the relatively

free algebra K(X)/Idgr (Un (K) of the variety of n-graded algebras var(Un (K)).

Proof. The map xij → Yij defines an epimorphism Φ: K(X) → Gn . Furthermore

Φ is actually an isomorphism. The reasoning is exactly the same as in the case of
the generic matrix algebra since our field K is infinite. Obviously Φ preserves the
grading and hence is a graded isomorphism.

We shall work in the algebra Gn instead of the relatively free algebra. We use
some ideas from [2] and [19].

Lemma 3.2. Let ms = ms (xi1 j1 , xi2 j2 , . . . , xik jk ), s = 1, 2, be two monomials in

K(X). Suppose that the matrices

M1 = m1 (Yi1 j1 , Yi2 j2 , . . . , Yik jk ) and M2 = m2 (Yi1 j1 , Yi2 j2 , . . . , Yik jk )

in Gn have the same nonzero entries in the same positions in their first rows. (In
other words, M1 − M2 has zeros in its first row.) Then M1 = M2 .

Proof. We know that the (unique) nonzero entry in the first row occurs in position
(1, r + 1), and the same entry occurs in M2 , at the same position (1, r + 1), for some
r, 0 ≤ r ≤ n − 1. One computes these entries as follows. For M1 , we have that
the (1, r + 1)st entry equals ya1 ,b1 ,c1 ya2 ,b2 ,c2 · · · yak ,bk ,ck where a1 = 1, b1 = i1 + 1,
c1 = j1 , and ct = jt for all t; thus one obtains the recurrence formula

at+1 = bt , bt+1 = at+1 + it+1 , ct+1 = jt+1 , 1 ≤ t ≤ k −1,

where a1 = 1, b1 = i1 + 1, c1 = j1 . Note that bt+1 = bt + it+1 . Furthermore, in

order to obtain a nonzero entry one needs bk ≤ n. But this is always the case if
bk = i1 +i2 +· · ·+ik +1 = r +1 ≤ n. Now observe that these recurrencies determine
uniquely the monomial M1 , and hence M1 = M2 .
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522 P. Koshlukov & A. Valenti

Corollary 3.1. Under the assumptions and in the notation of the previous lemma,
we have m1 − m2 ∈ Idgr (Un (K)).

Proof. The proof is straightforward since M1 −M2 = 0 in Gn which is the relatively

free n-graded algebra.

Using the properties of the relatively free graded algebra we give an alternative
proof of Theorem 2.1.
Proof of Theorem 2.1. Let I be the Tn -ideal generated by the polynomials (1)
and (2), and let J = Idgr (Un (K)). Since I ⊆ J, it suffices to prove that J ⊆ I.
Suppose on the contrary that there exists a multihomogeneous polynomial f ∈ J
and f 6∈ I. We work in the relatively free n-graded algebra K(X)/J = Gn and
choose f ∈ Gn of minimal degree, and among these f , choose one that is expressed
in the form f = α1 m1 + α2 m2 + · · · + αs ms for mt being distinct monomials in Gn ,
all αt 6= 0, αt ∈ K, and s the least possible. Hence s ≥ 1.
Suppose mt = mt (xi1 j1 , xi2 j2 , . . . , xik jk ). Then
s
X
m1 (Yi1 j1 , Yi2 j2 , . . . , Yik jk ) = βz mz (Yi1 j1 , Yi2 j2 , . . . , Yik jk )
z=2
where βz = −αz /α1 6= 0, z = 2, 3, . . . , s.
On the other hand we have that m1 (Yi1 j1 , Yi2 j2 , . . . , Yik jk ) 6= 0 and in the first
row of this matrix there will be some nonzero entry. This nonzero entry appears
on the right hand side as well. Say it comes from the monomial m2 . But then the
monomials m1 and m2 have the same nonzero entry in the same position on their
first rows, and hence m1 − m2 ∈ J, according to Corollary 3.1. But then m1 = m2 ,
and we reduce f to s − 1 monomials which contradicts the choice of s. Hence s = 0
and I = J.
We now prove that the monomials (3) are linearly independent modulo
Pt
Idgr (Un (K)). Suppose that the polynomial i=1 αi mi ∈ Idgr (Un (K)) where mi
are distinct monomials of the type (3), for 0 6= αi ∈ K. Since K is infinite we may
assume that the monomials mi are all multihomogeneous and of the same multide-
gree. One can express m1 in the following way: m1 = − ti=2 βi mi . Now substitute
P

the variables for the respective generic graded matrices; the first row of m1 will
contain some nonzero entry since otherwise m1 would belong to Idgr (Un (K)). The
same nonzero entry must appear in some of the monomials on the right hand side,
say in m2 . Hence m1 − m2 ∈ Idgr (Un (K)), and we reduce our linear combination to
t − 1 terms. Finally we obtain single monomial, but it cannot be a graded identity
for Un (K) since it is of weight < n, and it will contain in its first row some nonzero
entry when evaluated on Gn .

4. Applications
Here we apply the results of the previous section and describe some of the numerical
invariants of the Tn -ideal of Un (K). We start with one of the various kinds of
codimensions one may define for graded identities.
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Graded Identities for the Algebra of n × n Upper Triangular Matrices 523

Suppose that g1 h1 g2 h2 · · · gk hk is a multilinear monomial that is nonzero in

the relatively free n-graded algebra K(X)/Idgr (Un (K)). Let gi depend on the
0-variables x0j , and hi depend on some xzj , 1 ≤ z ≤ n − 1. Suppose further
that the number of variables x0j is s, and of the xzj is m − s. If gu depends on
tu variables, 1 ≤ u ≤ k, we fix these variables and we obtain (m − s)! monomials
of the desired type, permuting only the variables xzj . Now, dividing the variables
x0j into groups, ti in the ith group, we will obtain a multiple of the multinomial
coefficient
!
s!(m − s)! s
= (m − s)! .
(t1 !t2 ! · · · tk !) t1 !t2 ! · · · tk !
Now summing up over all such divisions of the x0j into k groups (some of them may
be empty) one gets that the mth graded codimension in the fixed variables equals
(m − s)!(m − s + 1)s . Of course we impose one further condition namely that the
weight of the monomial be less than n. Hence we proved the following proposition.

Proposition 4.1. Let m be a fixed positive integer and let x0j1 , x0j2 , . . . , x0js be
fixed. Suppose further that xz1 ,r1 , xz2 ,r2 , . . . , xzg ,rg , g = m − s, are fixed, 1 ≤ zi ≤
n−1 for all i and z1 +z2 +· · ·+zg ≤ n−1. Then the span of all multilinear monomials
on these variables in K(X)/Idgr (Un (K)) is of dimension (m − s)!(m − s + 1)s .
One can consider another kind of graded codimensions (see for example [3]).
We recall the definition of these codimensions that represent a direct generalization
of the ungraded case. Let m be fixed positive integer, and consider the (graded)
variables xij , 0 ≤ i ≤ n, 1 ≤ j ≤ m, in K(X). We consider the multilinear
monomials in these variables that are of the form
xii ,j1 xi2 ,j2 · · · xim ,jm (4)
where {j1 , j2 , . . . , jm } = {1, 2, . . . , m} and 0 ≤ it ≤ n − 1.
In other words, we do not admit repetitions of the second indices in the variables.
These monomials can be obtained by the usual multilinear monomials xj1 xj2 · · · xjm
by assigning all possible weights on the variables. It is straightforward that the
n (n)
span Pm of these monomials in K(X) is of dimension cm = nm m!. Denote by
Pmn
(A) the quotient Pm n
/(Pmn
∩ Idgr (A)) where A is an n-graded algebra, and set
(n) n (n)
cm (A) = dimK Pm (A). Next we compute the codimensions cm = cm (Un (K)).

Theorem 4.1. The codimensions cm equal

M
! !
X m n−1
cm = q!(q + 1)m−q
q=0
q q

where M = min{m, n − 1}.

n
Proof. Let f = g1 y1 g2 y2 · · · gk yk gk+1 ∈ Pm (Un (K)) be a nonzero multilinear
monomial in the variables xij , 0 ≤ i ≤ n−1, 1 ≤ j ≤ m where for i = 1, . . . , q+1, the
gi0 s are monomials (possibly empty) in the variables x0j only, and for r = 1, . . . , k,
December 2, 2003 17:17 WSPC/132-IJAC 00160

524 P. Koshlukov & A. Valenti

n
yr = xar ,br , with 1 ≤ ar ≤ n − 1, 1 ≤ br ≤ m. Since f ∈ Pm (Un (K)), we have to
impose a1 + a2 + · · · + aq ≤ n − 1.
Denote by An−1 (q) the number of q-tuples of positive integers a1 , a2 , . . . , aq such
that a1 + a2 + · · · + aq ≤ n − 1, and as Bn−1 (q) the number of such q-tuples with
a1 + a2 + · · · + aq = n − 1. Then obviously
An−1 (q) = Bn−1 (q) + Bn−2 (q) + · · · +
r−1
Bq (q). On the other hand, Br (q) = q−1 is the number of compositions of r into
q parts (see for example [1, p. 54]). Hence, using some elementary transformations
on binomial coefficients, we obtain
n−1 n−1
! !
X X q+r−1 n−1
An−1 (q) = Br (q) = = .
r=q r=q q−1 q

Now, for the indices br of yr = xar ,br , we have m!/(m − q)! = q! m

q choices (no
repetition!). Therefore there exist
! !
m n−1
q!
q q
nonzero monomials h in Gn with the properties described above.
Now let us consider the variables x0j . We have m − q choices for the index j
(again no repetitions!), and we split the set of all such x0j into q + 1 subsets (some
of them possibly empty). If we denote by C(q) the number of nonzero monomials f
of type (4) that can be represented as f = g1 y1 g2 y2 · · · gk yk gk+1 where g1 , . . . , gk+1
are monomials (possibly empty) in x0j , and yr = xar ,br with 1 ≤ ar ≤ n − 1, then
we have
n−1
XX q
cm = (C(k, q) − C(k − 1, q))
q=0 k=0

where
! !
m−q
m n−1
C(k, q) = (k + 1) q! = .
q q
Therefore
q
n−1
! !
XX m n−1
cm = q! = ((k + 1)m−q − k m−q )
q=1 k=0
q q

m
! !
X m n−1
= = q!(q + 1)m−q .
q=0
q q

Note that the binomial coefficient n−1

q equals 0 whenever q > n − 1, and this last
observation yields the formula for cm from the theorem.

As a consequence of Theorem 4.1, we can now compute the precise asymptotics

of the sequence of graded codimensions cm (Un (K)).
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Graded Identities for the Algebra of n × n Upper Triangular Matrices 525

Recall that if f (x) and g(x) are two functions of a natural argument then f (x)
(x)
and g(x) are asymptotically equal, and we write f (x) ' g(x) if limx→∞ fg(x) = 1.

Corollary 4.1. For all m,

1
cm (Un (K)) ' mn−1 nm .
nn−1
Proof. For M = min{m, n − 1}, we have
M
! ! !
X m n−1 m
cm (Un (K)) = q!(q + 1)m−q ' (n − 1)!nm−n+1
q=0
q q n−1
which in turn equals
m(m − 1) · · · (m − n + 2) 1
(n − 1)!nm−n+1 ' mn−1 nm−n+1 = n−1 mn−1 nm .
(n − 1)! n

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