On a Quantum Version of Pieri's Formula

12 Pages·2002·0.17 MB·English

On a Quantum Version of Pieri’s Formula
Alexander Postnikov
Department of Mathematics, M.I.T., Cambridge, MA 02139
E-mail address: apost@@math.mit.edu
March 23, 1997
Abstract
We give an algebro-combinatorial proof of a general version of Pieri’s for-
mula following the approach developed by Fomin and Kirillov in the paper
“Quadratic algebras, Dunkl elements, and Schubert calculus.” We prove sev-
eral conjectures posed in their paper. As a consequence, a new proof of clas-
sical Pieri’s formula for cohomology of complex flag manifolds, and that of its
analogue for quantum cohomology is obtained in this paper.
1 Introduction
The purpose of this paper is to investigate several consequences and generalizations
of quantum Monk’s formula from [FGP]. In our approach we follow Fomin and
Kirillov [FK], who constructed a certain quadratic algebra E equipped with a family
n
of pairwise commuting “Dunkl elements,” which generate a subalgebra canonically
isomorphic to the cohomology ring of complex flag manifold. They observed that
Pieri’s formula for the cohomology of flag manifolds can be deduced from a certain
identity, which conjecturally holds in E . In this paper a more general Pieri-type
n
formula is proved, which specializes to their conjecture in a particular case.
A quantum deformation of the algebra E was also given in [FK], along with
n
the conjecture that its subalgebra generated by the Dunkl elements is canonically
isomorphic to the (small) quantum cohomology of the flag manifold. This statement
is also a special case of our result.
Pieri’s formula for multiplication in the quantum cohomology ring of flag mani-
foldscanalsobeobtainedfromourformula(buttheoppositeisnottrue). Itsclassical
counterpart is the rule that was formulated by Lascoux and Schu¨tzenberger [LS] and
proved geometrically by Sottile [S] (see also [W] for a combinatorial proof). Pieri’s
formula for the quantum cohomology was recently proved by Ciocan-Fontanine [C2],
using nontrivial algebro-geometric techniques. By contrast, our proof is combina-
torial, and does not rely upon geometry at all—once (quantum) Monk’s formula is
given. Our proof seems to be new even in the classical case.
Key words and phrases. Flagmanifold, Monk’sformula, Pieri’sformula, quantumcohomology.
1
The rest of Introduction is devoted to a brief account of main notions and results
related to the classical as well as the quantum cohomology rings of complex flag
manifolds. For a more complete story, see [BGG], [FGP], [FP], [Ma], and bibliog-
raphy therein. Although many of the constructions below can be carried out in a
more general setup of an arbitrary semisimple Lie group, only the case of type A
n−1
is considered in this paper.
Let Fl denote the manifold of complete flags of subspaces in Cn. According
n
to classical Ehresmann’s result [E], the Schubert classes σ , indexed by the ele-
w
ments w of the symmetric group S , form an additive Z-basis of the cohomology
n
ring H∗(Fl ,Z) of the flag manifold.
n
The multiplicative structure of H∗(Fl ,Z) can be recovered from Borel’s theo-
n
rem [B]. Let s be the element of S that transposes i and j. Also let s = s ,
ij n i ii+1
1 ≤ i ≤ n−1, be the Coxeter generators of S . Borel’s theorem says that H∗(Fl ,Z)
n n
is canonically isomorphic, as a graded algebra, to the quotient
Z[x ,x ,...,x ]/he ,e ,...,e i, (1)
1 2 n 1 2 n
where e = e (x ,x ,...,x ) denotes the kth elementary symmetric polynomial, for
k k 1 2 n
k = 1,2,...,n; and he ,...,e i is the ideal generated by the e . The isomorphism
1 n k
is given by explicitly specifying
x +x +···+x 7−→ σ , m = 1,2,...,n−1.
1 2 m sm
A way to relate these two descriptions of the cohomology ring H∗(Fl ,Z) was
n
found by Bernstein, Gelfand, and Gelfand [BGG] and Demazure [D]. Lascoux and
Schu¨tzenberger [LS] then constructed the Schubert polynomials, whose images in the
quotient (1) represent the Schubert classes σ .
w
Recently, attention has been drawn to the (small) quantum cohomology ring
QH∗(Fl ,Z) of the flag manifold. We will not give here the definition of quantum
n
cohomology (see e.g. [FP]), but we mention that structure constants of quantum co-
homology are 3-point Gromov-Witten invariants, which count the numbers of certain
rational curves and play a role in enumerative algebraic geometry.
As a vector space, the quantum cohomology of Fl is essentially the same as the
n
usual cohomology. More precisely,
QH∗(Fl ,Z) ∼= H∗(Fl ,Z)⊗Z[q ,...,q ].
n n 1 n−1
However, the multiplicative structure in QH∗(Fl ,Z) is different.
n
AquantumanalogueofBorel’stheoremwassuggestedbyGiventalandKim[GK],
and then justified by Kim [K] and Ciocan-Fontanine [C1]. Let E ,E ,...,E ∈
1 2 n
Z[x ,...,x ;q ,...,q ] be the nonidentity coefficients of the characteristic polyno-
1 n 1 n−1
mial of the matrix
 
x q 0 ··· 0
1 1
 −1 x q ··· 0 
2 2
 
 0 −1 x3 ··· 0  . (2)
 
 ... ... ... ... ... 
 
0 0 0 ··· x
n
2
The E are certain q-deformations of the elementary symmetric polynomials e . If
k k
q = ··· = q = 0 then E specializes to e .
1 n−1 k k
Givental, Kim, and Ciocan-Fontanine showed that the quantum cohomology
ring QH∗(Fl ,Z) is canonically isomorphic to the quotient
n
Z[x ,...,x ;q ,...,q ]/hE ,E ,...,E i. (3)
1 n 1 n−1 1 2 n
Just as in the classical case, the isomorphism is given by specifying
x +x +···+x 7−→ σ , m = 1,2,...,n−1. (4)
1 2 m sm
An important problem is to find the expansion of the quantum product σ ∗σ
u w
of two Schubert classes in the basis of Schubert classes, where “∗” denotes the mul-
tiplication in the quantum cohomology ring.
This problem was solved, or at least reduced to combinatorics, in [FGP]. In
that paper we gave a quantum analogue of the Bernstein-Gelfand-Gelfand theo-
rem and the corresponding deformation of Schubert polynomials of Lascoux and
Schu¨tzenberger. We also proved there a quantum Monk’s formula, which generalizes
the classical Monk’s result [Mo].
Let us denote q = q q ···q , for i < j.
ij i i+1 j−1
Theorem 1.1 (Quantum Monk’s formula) [FGP, Theorem 1.3] For w ∈ S and
n
1 ≤ m < n, the quantum product of Schubert classes σ and σ is given by
sm w
X X
σ ∗σ = σ + q σ , (5)
sm w wsab cd wscd
where the first sum is over all transpositions s such that a ≤ m < b and ‘(ws ) =
ab ab
‘(w)+1, and the second sum is over all transpositions s such that c ≤ m < d and
cd
‘(ws ) = ‘(w)−‘(s ) = ‘(w)−2(d−c)+1.
cd cd
The formula (5) unambiguously determines the multiplicative structure of the
quantum cohomology ring QH∗(Fl ,Z) with respect to the basis of Schubert classes,
n
since this ring is generated by the 2-dimensional classes σ .
sr
Asanexample, wededucearuleforthequantumproductofanyclassσ withthe
w
classσ ,wherec(k,m) = s s ···s (Corollary4.3). Themainresultof
c(k,m) m−k+1 m−k+2 m
ourpaperisevenmoregeneralstatement(Theorem3.1)thatwecall“quantumPieri’s
formula.” It is formulated in the language of the construction for the cohomology
of Fl highlighted by Fomin and Kirillov. We were able to extend some of their
n
results and provide a proof to the following conjectures from [FK]: Conjecture 11.1,
Conjecture 13.4, and Conjecture 15.1.
Acknowledgments. I am grateful to Sergey Fomin for introducing me to his
paper with Anatol Kirillov [FK].
3
2 Definitions
Let us recall the definition of the quadratic algebra Ep given by Fomin and Kir-
n
illov [FK, Section 15]. (Their notation is slightly different from ours.) The algebra
Ep is generated over Z by the elements τ and p , i,j ∈ {1,2,...,n}, subject to the
n ij ij
following relations:
τ = −τ , τ = 0, (6)
ij ji ii
τ2 = p , (7)
ij ij
τ τ +τ τ +τ τ = 0, (8)
ij jk jk ki ki ij
[p ,p ] = [p ,τ ] = 0, for any i,j,k, and l, (9)
ij kl ij kl
[τ ,τ ] = 0, for any distinct i,j,k, and l. (10)
ij kl
Here [a,b] = ab − ba is the usual commutator. Remark that the generator τ was
ij
denoted by [ij] in [FK]. It follows from (6) and (7) that p = p and p = 0.
ij ji ii
Thecommutingelementsp canbeviewedasformalparameters. ThequotientE
ij n
of the algebra Ep modulo the ideal generated by the p was the main object of study
n ij
in [FK]. Also an algebra Eq was introduced in that paper. It can be defined as the
n
quotient of Ep by the ideal generated by the p with |i−j| ≥ 2. The image of p
n ij ii+1
in Eq is denoted q .
n i
Following [FK, Section 5], define the “Dunkl elements” θ , i = 1,...,n, in the
i
algebra Ep by
n
n
X
θ = τ . (11)
i ij
j=1
The following important property of these elements is not hard to deduce from
the relations (6)–(10).
Lemma 2.1 [FK,Corollary5.2andSection15] The elements θ ,θ ,...,θ commute
1 2 n
pairwise.
Let x ,x ,...,x be a set of commuting variables, and let p be a shorthand for
1 2 n
the collection of p ’s. For a subset I = {i ,...,i } in {1,2...,n}, we denote by
ij 1 m
x the collection of variables x ,...,x . Define the quantum elementary symmetric
I i1 im
polynomial (cf. [FGP, Section 3.2] or [FK, Section 15])
E (x ;p) = E (x ,x ,...,x ;p) (12)
k I k i1 i2 im
by the following recursive formulas:
E (x ,x ,...,x ;p) = 1, (13)
0 i1 i2 im
E (x ,x ,...,x ;p) = E (x ,x ,...,x ;p)
k i1 i2 im k i1 i2 im−1
+ E (x ,x ,...,x ;p)x (14)
k−1 i1 i2 im−1 im
m−1
X
+ E (x ,...,x ,...,x ;p)p ,
k−2 i1 cir im−1 irim
r=1
4
where the notation x means that the corresponding term is omitted.
cir
The polynomial E (x ;p) is symmetric in the sense that it is invariant under the
k I
simultaneous action of S on the variables x and the p . One can directly verify
m ia iaib
from (13) and (14) that
E (x ,x ,...,x ;p) = x +x +···+x ,
1 i1 i2 im i1 i2 im
X
E (x ,x ,...,x ;p) = (x x +p ).
2 i1 i2 im ia ib iaib
1≤a<b≤m
The polynomials E (x ;p) have the following elementary monomer-dimer inter-
k I
pretation (cf. [FGP, Section 3.2]). A partial matching on the vertex set I is a un-
ordered collection of “dimers” {a ,b },{a ,b },... and “monomers” {c },{c },...
1 1 2 2 1 2
such that all a ,b ,c are distinct elements in I. The weight of a matching is the
i j k
productp p ···x x ···. ThenE (x ;p)isthesumofweightsofallmatchings
a1b1 a2b2 c1 c2 k I
which cover exactly k vertices of I.
For example, we have
E (x ,x ,x ,x ;p) = x x x +x x x +x x x +x x x
3 1 2 3 4 1 2 3 1 2 4 1 3 4 2 3 4
+p (x +x )+p (x +x )+p (x +x )
12 3 4 13 2 4 14 2 3
+p (x +x )+p (x +x )+p (x +x ).
23 1 4 24 1 3 34 1 2
Specializing p = 0, one obtains E (x ;0) = e (x ), the usual elementary sym-
ij k I k I
metric polynomial. Assume that p = q , i = 1,2,...,n − 1, and p = 0, for
ii+1 i ij
|i−j| ≥ 2. Then the polynomial E (x ,...,x ;q) is the quantum elementary poly-
k 1 n
nomial E , which is a coefficient of the characteristic polynomial of the matrix (2).
k
Here and below the letter q stands for the collection of q ,q ,...,q .
1 2 n−1
3 Main result
For a subset I = {i ,...,i } in {1,2,...,n}, let θ denote the collection of the
1 m I
elements θ ,...,θ , and let E (θ ;p) = E (θ ,...,θ ;p) denote the result of sub-
i1 im k I k i1 im
stituting the Dunkl elements (11) in place of the corresponding x in (12). This
i
substitution is well defined, due to Lemma 2.1. Our main result can be stated as
follows:
Theorem 3.1 (Quantum Pieri’s formula) Let I be a subset in {1,2,...,n}, and let
J = {1,2,...,n}\I. Then, for k ≥ 1, we have in the algebra Ep:
n
X
E (θ ;p) = τ τ ···τ , (15)
k I a1b1 a2b2 akbk
where the sum is over all sequences a ,...,a ,b ,...,b such that (i) a ∈ I, b ∈ J,
1 k 1 k j j
for j = 1,...,k; (ii) the a ,...,a are distinct; (iii) b ≤ ··· ≤ b .
1 k 1 k
Note that all terms in (15) involving the p cancel each other.
ij
5
The proof of Theorem 3.1 will be given in Section 5. In the rest of this section
we summarize several corollaries of Theorem 3.1.
First of all, let us note that specializing p = 0 in Theorem 3.1 results in Con-
ij
jecture 11.1 from [FK]. Conjecture 15.1 is also a consequence of our result.
Corollary 3.2 [FK, Conjecture 15.1] For k = 1,2,...,n, the following relation in
the algebra Ep holds
n
E (θ ,θ ,...,θ ;p) = 0.
k 1 2 n
Proof. In this case, the sum in (15) is over the empty set. (cid:3)
Define a Z[p]-linear homomorphism π by
π : Z[x ,x ,...,x ;p] −→ Ep
1 2 n n
π : x 7−→ θ .
i i
Corollary 3.3 The kernel of π is generated over Z[p] by
E (x ,x ,...,x ;p), k = 1,2,...,n. (16)
k 1 2 n
Proof. All elements (16) map to zero, due to Corollary 3.2. The statement now
follows from a dimension argument (cf. [FK, Section 7]). (cid:3)
In particular, we can define a homomorphism π¯ by
π¯ : Z[x ,...,x ] −→ E ,
1 n n
¯
π¯ : x 7−→ θ ,
i i
¯
where θ is the image in E of the element θ .
i n i
Corollary 3.4 [FK, Theorem 7.1] The kernel of π¯ is generated by the elementary
symmetric polynomials
e (x ,x ,...,x ), k = 1,2,...,n.
k 1 2 n
¯
Thus the subalgebra in E generated by the θ is canonically isomorphic to the coho-
n i
mology of Fl , which is isomorphic to the quotient (1).
n
Likewise, let θˆ be the image in Eq of the element θ , and let πˆ be the Z[q]-linear
i n i
homomorphism defined by
πˆ : Z[x ,...,x ;q] −→ Eq,
1 n n
ˆ
πˆ : x 7−→ θ .
i i
Corollary 3.5 [FK, Conjecture 13.4] The kernel of the homomorphism πˆ is gener-
ated over Z[q] by
E (x ,x ,...,x ;q), k = 1,2,...,n.
k 1 2 n
Thus the subalgebra in Eq generated over Z[q] by the θˆ is canonically isomorphic to
n i
the quantum cohomology of Fl , the latter being isomorphic to the quotient (3).
n
6
4 Action on the quantum cohomology
Recall that s is the transposition of i and j in S , s = s is a Coxeter generator,
ij n i ii+1
and q = q q ···q , for i < j.
ij i i+1 j−1
Let us define the Z[q]-linear operators t , 1 ≤ i < j ≤ n, acting on the quantum
ij
cohomology ring QH∗(Fl ,Z) by
n

σ if ‘(ws ) = ‘(w)+1,
 wsij ij
t (σ ) = q σ if ‘(ws ) = ‘(w)−2(j −i)+1, (17)
ij w ij wsij ij

 0 otherwise.
By convention, t = −t , for i > j, and t = 0.
ij ji ii
Quantum Monk’s formula (Theorem 1.1) can be stated as saying that the quan-
tum product of σ and σ is equal to
sm w
X
σ ∗σ = t (σ ).
sm w ab w
a≤m<b
The relation between the algebra Eq and quantum cohomology of Fl is justified
n n
by the following lemma, which is proved by a direct verification.
Lemma 4.1 [FK, Proposition 12.3] The operators t given by (17) satisfy the re-
ij
lations (6)–(10) with τ replaced by t , p = q , and p = 0, for |i−j| ≥ 2,
ij ij ii+1 i ij
Thus the algebra Eq acts on QH∗(Fl ,Z) by Z[q]-linear transformations
n n
τ : σ 7−→ t (σ ).
ij w ij w
ˆ
Monk’s formula is also equivalent to the claim that the Dunkl element θ acts on
i
the quantum cohomology of Fl as the operator of multiplication by x , the latter is
n i
defined via the isomorphism (4).
Let us denote c(k,m) = s s ···s and r(k,m) = s s ···s .
m−k+1 m−k+2 m m+k−1 m+k−1 m
Thesearetwocyclicpermutationssuchthatc(k,m) = (m−k+1,m−k+2,...,m+1)
and r(k,m) = (m+k,m+k −1,...,m).
The following statement was geometrically proved in [C1] (cf. also [FGP]). For
the reader’s convenience and for consistency we show how to deduce it directly from
Monk’s formula.
Lemma 4.2 The coset of the polynomial E (x ,...,x ;q) in the quotient ring (3)
k 1 m
corresponds to the Schubert class σ under the isomorphism (4). Analogously,
c(k,m)
the coset of the polynomial E (x ,x ,...,x ) corresponds to the class σ .
k m+1 m+2 n r(k,m)
Proof. By (4) and (14), it is enough to check that
σ = σ +(σ −σ )∗σ +q σ .
c(k,m) c(k,m−1) sm sm−1 c(k−1,m−1) m−1 c(k−2,m−2)
This identity immediately follows from Monk’s formula:
X X
(σ −σ )∗σ = ( t − t )(σ ).
sm sm−1 c(k−1,m−1) mb am c(k−1,m−1)
b>m a<m
7
The claim about σ can be proved using a symmetric argument. (cid:3)
r(k,m)
It is clear now that Theorem 3.1 implies the following statement. This statement,
though in a different form, was proved in [C2].
Corollary 4.3 (Quantum Pieri’s formulas: QH∗ version) For w ∈ S and 0 ≤ k ≤
n
m < n, the product of Schubert classes σ and σ in the quantum cohomology
c(k,m) w
ring QH∗(Fl ,Z) is given by the formula
n
X
σ ∗σ = t t ···t (σ ), (18)
c(k,m) w a1b1 a2b2 akbm w
where the sum is over a ,...,a ,b ,...,b such that (i) 1 ≤ a ≤ m < b < n for
1 k 1 k j j
j = 1,...,k; (ii) the a ,...,a are distinct; (iii) b ≤ ··· ≤ b .
1 k 1 k
Likewise, the quantum product of Schubert classes σ and σ is given by the
r(k,m) w
formula
X
σ ∗σ = t t ···t (σ ), (19)
r(k,m) w c1d1 c2d2 ckdk w
where the sum is over c ,...,c ,b ,...,d such that (i) 1 ≤ c ≤ m < d < n for
1 k 1 k j j
j = 1,...,k; (ii) c ≤ ··· ≤ c ; (iii) the d ,...,d are distinct.
1 k 1 k
Note that Corollary 4.3 does not imply Theorem 3.1 (or even its weaker form
for Eq), since the representation τ 7→ t of Eq in the quantum cohomology is not
n ij ij n
exact.
5 Proof of Theorem 3.1
For a subset I in {1,2,...,n}, let Ee (I) denote the expression in the right-hand side
k
of (15). By convention, Ee (I) = 1. For k = 1, Theorem 3.1 says that
0
n
XX XX
Ee (I) = τ = τ = E (θ ;p),
1 ij ij 1 I
i∈I j6∈I i∈I j=1
which is obvious by (6).
It suffices to verify that the Ee (I) satisfy the defining relation (14). Then the
k
claim E (θ ;p) = Ee (I) will follow by induction on k. Specifically, we have to
k I k
demonstrate that
X
Ee (I ∪{j}) = Ee (I)+Ee (I)θ + Ee (I \{i})p , (20)
k k k−1 j k−2 ij
i∈I
where I ⊂ {1,2,...,n} and j 6∈ I. To do this we need some extra notation. For a
subset L = {l ,l ,...,l } and r 6∈ L, denote
1 2 m
X
hhL | rii = τ τ ···τ ,
u1r u2r umr
where the sum is over all permutations u ,u ,...,u of l ,l ,...,l .
1 2 m 1 2 m
8
For I and j as in (20), let J = {1,2,...,n} \ I = {j ,j ,...,j } with j = j.
1 2 d 1
Then the first term in the right-hand side of (20) can be written in the form
X
Ee (I) = hhI | j ii hhI | j ii···hhI | j ii, (21)
k 1 1 2 2 d d
I1...Id⊂kI
where the notation I ...I ⊂ I means that the sum is over all pairwise disjoint
1 d k
P
(possibly empty) subsets I ,I ,...,I of I such that |I | = k. Let
1 2 d s s
Ee (I) = A +A , (22)
k 1 2
where A is the sum of terms in (21) with I = ∅ and A is the sum of terms with
1 1 2
I 6= ∅. Likewise, we can split the left-hand side of (20) into two parts:
1
X
Ee (I ∪{j}) = hhI0 | j ii hhI0 | j ii···hhI0 | j ii
k 2 2 3 3 d d
I20···Id0⊂kI∪{j} (23)
= B +B ,
1 2
where B is the sum of the terms such that j 6∈ I0 ∪···∪I0, and B is the sum of
1 2 d 2
terms with j ∈ I0 ∪ ··· ∪ I0. We also split the second term in the right-hand side
2 d
of (20) into 3 summands:
X X
Ee (I)θ = hhI00 | j ii···hhI00 | j ii τ
k−1 j 1 1 d d js
I100...Id00⊂k−1I s6=j (24)
= C +C +C ,
1 2 3
where C is the sum of terms with s ∈ I\(I00∪I00∪···∪I00); C is the sum of terms
1 1 2 d 2
with s ∈ I00 ∪I00 ∪···∪I00 ∪J; and C is the sum of terms with s ∈ I00.
2 3 d 3 1
It is immediate from the definitions that A = B . It is also not hard to verify
1 1
that A +C = 0, since for I 6= ∅
2 1 1
X
hhI | j ii = hhI \{i} | j ii τ .
1 1 1 1 ij1
i∈I1
To prove the identity (20), it thus suffices to demonstrate that
B = C , (25)
2 2
X
C + Ee (I \{i})p = 0. (26)
3 k−2 ij
i∈I
The following lemma implies the formula (25).
Lemma 5.1 For any subset K in {1,2,...,n} and j, l 6∈ K, we have
X X
hhK ∪{j} | lii = hhL | lii hhK \L | jii τ . (27)
js
L⊂K s∈L∪{l}
9
Indeed, let T = hhI0 | j ii···hhI0 | j ii be a term of B . Then j ∈ J0 for some r.
2 2 d d 2 r
By Lemma 5.1, T is equal the sum of all terms hhI00 | j ii···hhI00 | j ii τ in C with
1 1 d d js 2
fixed I00 = I0 for all u 6= r such that s ∈ I00∪{j } and the subsets I00∪I00 = I0 \{j}.
u u r r 1 r r
Thus B = C .
2 2
Proof of Lemma 5.1. Induction on |K|. For K = ∅, the both sides of (27) are
equal to τ . For |K| ≥ 1, the right-hand side of (27) is equal
jl
X X
hhL | lii hhK \L | jii τ
js
L⊂K s∈L∪{l}
 
X X X X
=  hhL | lii τij hhK \L\{i} | jii τjs+hhK | lii τjs
L&K i∈K\L s∈L∪{l} s∈K∪{l}
X X
= τ hh(K \{i})∪{j} | lii+hhK | lii τ
ij js
i∈K s∈K∪{l}
= hhK ∪{j} | lii.
The second equality is valid by induction hypothesis; the remaining equalities follow
from (8) and (10). (cid:3)
Using a similar argument to the one after Lemma 5.1, one can derive the for-
mula (26) from the following lemma:
Lemma 5.2 For any subset K in {1,2,...,n} and j 6∈ K, we have
 
X X
hhK | jii τjs + hhL | sii hhK \L\{s} | jiipjs = 0.
s∈K L⊂K\{s}
This statement, in turn, is obtained from the following “quantum analogue” of
Lemma 7.2 from [FK]. Its proof is a straightforward extension.
Lemma 5.3 For i,u ,u ,...,u ∈ {1,...,n}, we have in the algebra Ep
1 2 m n
m
X
τ τ ···τ τ τ ···τ
iur iur+1 ium iu1 iu2 iur
r=1 (28)
m
X
= p τ τ ···τ τ τ ···τ ,
iur urur+1 urur+2 urum uru1 uru2 urur−1
r=1
where, by convention, the index u is identified with u .
m+1 1
Proof. Induction on m. The base of induction, for m = 1, is easily established
by (7): τ τ = p . Assume that m > 1. Applying (8) and (10) to the left-hand
iu1 iu1 iu1
10