This paper introduces two new formulas for Schubert polynomials using Gröbner-Shirshov bases, explores their combinatorial properties, and provides algorithms for calculating their structure constants, enhancing computational methods in algebraic combinatorics.
Contribution
It presents novel formulas for Schubert polynomials based on Gröbner-Shirshov bases and develops algorithms for their structure constants, including one utilizing Monk's formula.
Findings
01
Two formulas for Schubert polynomials involving only nonnegative monomials
02
Proved combinatorial properties of Schubert polynomials
03
Developed algorithms for calculating structure constants
Abstract
By applying a Gr\"{o}bner-Shirshov basis of the symmetric group Sn, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk's formula.
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On formulas and some combinatorial properties of Schubert Polynomials111Supported by the NSFC of China (11571121) and the Science and Technology Program of Guangzhou (201707010137).
Zerui Zhang222Supported by the Innovation Project of Graduate School of South China Normal University., Yuqun
Chen333Corresponding author.
School of Mathematical Sciences, South China Normal
University
Abstract: By applying a Gröbner-Shirshov basis of the symmetric group Sn, we give two formulas for
Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials.
As applications,
we give two algorithms to calculate the structure constants for Schubert polynomials, one of which
depends on Monk’s formula.
Schubert polynomials were introduced by I.N. Bernstein, I.M. Gelfand and S.I. Gelfand [3] and Michel Demazure [7] (in
the context of arbitrary root systems) and were extensively developed by Alain Lascoux, Marcel-Paul
Schützenberger [11, 12]. There are lots of other papers about approaches for the computations of
Schubert polynomials, for example, Sara C. Billey, William Jockusch and
Richard P. Stanley [4], Allen Knutson, Ezra Miller [10],
Rudolf Winkel [19, 21], Sergey Fomin, Richard P. Stanley [8], for more details, see [21].
For any u in the symmetric group Sn, u is associated with a Schubert polynomial, denoted by Su. It is well known that
there are identities
[TABLE]
where the structure constants cu,vw are some nonnegative integers [14, 9]. Though there are algorithms to calculate cu,vw, there are still no combinatorial proof. It is an open problem
to find a combinatorial rule for these coefficients. What people know about these coefficients are limited. One case where an explicit formula is known is
Monk’s Formula [16]. There are a lot of research on the multiplication of Schubert polynomials, for example, see [1, 2, 13, 18, 20].
The aim of this paper is to give another two formulas for Schubert polynomials, to establish some combinatorial properties for Schubert polynomials and to find algorithms
to calculate the structure constants for Schubert polynomials.
Our approach is algebraic, which is based on a Gröbner-Shirshov basis of the symmetric group Sn
defined by s1,…,sn−1 subjected to the relations : si2=1,
sisi+1si=si+1sisi+1, sisj=sjsi (j>i+1).
In section 2 we give two formulas for Schubert polynomials. The notations are introduced before the corresponding theorems. The formulas take the form:
where the summation is over all the Tn−1,…,T1 such that u(Tn−1,…,Tl) is defined for
any 1≤l≤n−1.
One of the corollaries of these two formulas is the well-known fact that the coefficients of monomials of any Schubert polynomial are nonnegative [14, 9]. By
Theorem 2, we develop some combinatorial properties of Schubert polynomials in Section 3.
We analysis how to write down the leading monomial of ∂uSw0n with
respect to some order, where w0n is the longest word in Sn. We also offer an algorithm to find a u∈Sn (n large enough) such that
the leading monomial of ∂uSw0n is a given commutative word. We show
that
∂tSu=∂tSu
if degxtSu>degxt+1Su,
where for any polynomial f, fˉ means the leading monomial of f.
In section 4, we show how the properties and formulas we established can
be applied to calculate the structure constants. We also explain how to apply Monk’s formula to the calculation of the structure constants.
As results,
we give two algorithms to calculate the structure constants for the multiplications of Schubert polynomials.
2 Two formulas for Schubert polynomials
The symmetric group, Sn, consists of all bijections from {1,2,…,n} to
itself using composition as the multiplication [17]. It is well known that Sn can be
defined by generators s1,…,sn−1 with relations: si2=1, sisi+1si=si+1sisi+1, sisj=sjsi (j>i+1),
where si corresponds to the adjacent transposition (i,i+1)∈Sn for each 1≤i≤n−1.
Let S={si∣1≤i≤n−1} (s1<s2<⋯<sn−1) and S∗ be a free monoid generated by S.
We define the degree lexicographic order on S∗ by the following:
for any u=si1si2⋯sip,v=sj1sj2⋯sjq∈S∗, where each sil,sjt∈S,
[TABLE]
We also define the degree of u, denoted by ∣u∣, to be p if u=si1si2⋯sip∈S∗.
Using the theory of Gröbner-Shirshov bases theory of associative algebras [5],
we know that, under the above definition of Sn by generators and relations, Sn has a Gröbner-Shirshov basis,
with respect to degree lexicographic order on S∗, as follows:
(1)
si2=1, 1≤i≤n−1;
2. (2)
sisj=sjsi, i>j+1,1≤j<i≤n−1;
3. (3)
si,jsi=si−1si,j, i>j, where
si,j is defined to be sisi−1⋯sj if j≤i and 1 otherwise (1 means the identity element of Sn).
Then the follow set
[TABLE]
consists of normal forms of elements of Sn.
For example, s3,2s5,3∈Bsn (n≥6).
For some historical reason, we also call Gröbner basis as Gröbner-Shirshov basis for noncommutative cases, for more details, see a survey [6].
For any u∈S∗, we call u
a reduced word if for any v∈S∗ with u=v∈Sn, then ∣u∣≤∣v∣. For any u∈S∗, let [u]∈Bsn be the normal form of u with respect to the above Gröbner-Shirshov basis. Since we use degree lexicographic order, we have that u is a reduced word if and only if ∣u∣=∣[u]∣. In other words, we can apply only relations (2) and (3)
of the Gröbner-Shirshov basis of Sn to rewrite u to the normal form [u].
Moreover, the length of u, denoted by l(u), is defined to be ∣[u]∣. For example, u=s5s4s3s5s4 is reduced, for u can be rewritten to s4s3s5s4s3 and the latter is a normal form.
From now on, by u∈Sn, we always assume that u∈Bsn unless otherwise specified.
For any 0<n∈N, where N is the set of nonnegative integers, define a group homorphism σn:Sn⟶Sn+1,
induced by si↦si,1≤i≤n−1. It is clear that σn is an embedding, i.e., Sn⊂Sn+1. So we
can define S∞=∪n≥1Sn.
Let Z[x1,…,xn] be the free commutative algebra generated
by {x1,…,xn} over Z, where Z is the
ring of integer numbers. For any polynomial f∈Z[x1,…,xn], for any i between 1 and n−1,
denote by sif the result of interchanging xi and xi+1 in f. Define
the divided difference operators [9] ∂i on the
polynomial ring Z[x1,…,xn] by the rule:
[TABLE]
Since f−sif is divisible by xi−xi+1, we know that ∂i(f) is still a polynomial in Z[x1,…,xn].
It follows immediately from the definition that for any f,g∈Z[x1,…,xn],
[TABLE]
In particular, if f=sif, then
∂i(fg)=f⋅∂i(g).
Define ∂i0=id (the identity map). Let p∈{0,1}. Then
[TABLE]
By the definition of ∂t we have that
[TABLE]
Define
[TABLE]
Denote by ⊕b∈BxZb the free Z-module
with Z-basis Bx.
It follows that for any polynomial f∈⊕b∈BxZb, we have ∂tf∈⊕b∈BxZb for any 1≤t≤n−1.
The divided difference operators ∂i’s satisfy the
nilCoxeter relations [8]:
R={∂i2=0, 1≤i≤n−1; ∂i∂i+1∂i=∂i+1∂i∂i+1, 1≤i≤n−2; ∂i∂j=∂j∂i, j>i+1}.
It is easy to see that the following relations
(i)
∂i2=0, 1≤i≤n−1,
2. (ii)
∂i∂j=∂j∂i, i>j+1,1≤j<i≤n−1,
3. (iii)
∂i,j∂i=∂i−1∂i,j, i>j, where
∂i,j is defined to be ∂i∂i−1⋯∂j if j≤i and id otherwise,
form a Gröbner-Shirshov basis of the associative algebra Z⟨∂1,∂2,…,∂n−1∣R⟩ generated by {∂i∣1≤i≤n−1} with relations R over Z.
This algebra is called nilCoxeter algebra, and is denoted by NCn.
It follows that a Z-basis of this algebra is
[TABLE]
For any word u=si1si2⋯sit∈S∗, define
[TABLE]
It follows that if u∈Sn⊂S∞ is not reduced, then by applying (2) and (3) of the Gröbner-Shirshov basis of Sn (if necessary),
u can be rewritten to v1sisiv2 for some i. By applying (ii)
and (iii) of the Gröbner-Shirshov basis of NCn, we
have ∂u=∂v1sisiv2=0. By similar reasoning, if u and v are
two reduced words with u=v∈Sn, then ∂u=∂v.
Let w0n=s1,1s2,1⋯sn−1,1∈Sn. For any w∈Sn, define
the Schubert polynomial corresponding to w as
[TABLE]
where [w−1w0n] is the normal form of w−1w0n with respect to the above Gröbner-Shirshov basis of Sn.
In particular, Sw0n=x1n−1x2n−2⋯xn−22xn−1 and ∂uSw0n is the Schubert polynomial corresponding to w0nu−1.
It is easy to see that, if w∈Sn⊆Sn+1, then [w−1w0n+1]=[w−1w0n]sn,1, and thus
Sw=∂[w−1w0n](Sw0n)=∂[w−1w0n+1](Sw0n+1).
It is an not obvious fact that the coefficients
of ∂uSw0n are nonnegative integers, see [14]. We will give two simple formulas for ∂uSw0n in the sequel, by either of which follows that the coefficients of ∂uSw0n
are nonnegative.
Define
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
For any unary linear operator δ1,δ2,…,δn−1 from
Z[x1,x2,…,xn] to Z[x1,x2,…,xn], define
[TABLE]
Lemma 2.1**.**
For any t∈[1,n], δi∈{id,∂i,si}(t≤i≤n−1),
δn∈{id,∂i}, we have
[TABLE]
where pt,qj∈{0,1} for any t≤j≤n.
Proof.
Induction on t. If t=n, it is clear.
Suppose t≤n,
[δtxt⋯δnxn]R=ptxtqtxt+1qt+1⋯xnqn. Then
[δt−1xt−1δtxt⋯δnxn]R=ptδt−1(xt−1xtqtxt+1qt+1⋯xnqn).
If δt−1=id, it is clear. If δt−1=st−1, then
[δt−1xt−1δtxt⋯δnxn]R=ptxt−1qtxtxt+1qt+1⋯xnqn.
If δt−1=∂t−1, then [δt−1xt−1δtxt⋯δnxn]R=(1−qt)ptxt+1qt+1⋯xnqn=(1−qt)ptxt−10xt0xt+1qt+1⋯xnqn.
Lemma 2.2**.**
For any f∈Z[x1,…,xn], i,j,k∈{1,2,⋯,n−1}, we have
(1)
If k>i or k<j or j>i, then
[TABLE]
2. (2)
If i≥k≥j, then
[TABLE]
Proof.
(1) is easy, so we just need to prove (2). Since
∂i,j(X[1,k]f)=∂i,k(X[1,k]∂k−1,jf), it is sufficient to
show that ∂i,k(X[1,k]f)=k≤t≤i+1∑X[1,k−1]xi+1P(t>i)∂t∂i,kf.
Induction on i−k.
If i−k=0, then i=k and
[TABLE]
Suppose the result holds for any i−k<l. Let i−k=l≥1. Then
[TABLE]
Define s0=s0,i=1 for any i≥1. Given n∈Z (n>1),
for any u=s1,i1⋯sn−1,in−1∈Sn, for any subset
Jn−1 of {l∈[1,n−2]∣il≤l},
define
[TABLE]
[TABLE]
[TABLE]
where rq=q if q∈Jn−1, rq=q−1 if q∈[1,n−1]∖Jn−1, [sr1,i1sr2,i2⋯srn−1,in−1]∈Sn−1 is the normal form of sr1,i1sr2,i2⋯srn−1,in−1. Remind that for any w∈S∗, w is reduced if and only if w can be rewritten to its normal form by applying (2) and (3) of the Gröbner-Shirshov basis of Sn.
We proceed to define u(Jn−1,…,Jn−l−1) by induction on l.
Suppose that u(Jn−1,…,Jn−l) has been defined.
If u(Jn−1,…,Jn−l) is undefined, then
Jn−l−1, XJn−l−1u(Jn−1,…,Jn−l) and
u(Jn−1,…,Jn−l−1) are undefined. Otherwise, say
u(Jn−1,…,Jn−l)=s1,j1⋯sn−l−1,jn−l−1∈Sn−l. Then for any subset
Jn−l−1 of {k∈[1,n−l−2]∣jk≤k}, we can define
XJn−l−1u(Jn−1,…,Jn−l) and then define
u(Jn−1,…,Jn−l,Jn−l−1) to be (u(Jn−1,…,Jn−l))(Jn−l−1).
Note that Jn−1 depends on u, Jn−2 depends on u(Jn−1), and so on.
Theorem 1**.**
For any u=s1,i1⋯sn−1,in−1∈Sn(n≥2), we have
[TABLE]
where the summation is over all the Jn−1,…,J1 such that u(Jn−1,…,Jl) is defined for
any 1≤l≤n−1.
Proof.
Induction on n.
If n=2, then u=s1,i1, i1= 1 or 2. For either case, we have J1=∅, u(J1)=1, ∂uSw02=∂u(x1)=X[1,i1−1]=XJ1u.
If n≥3,
we first show that for any u=st,itst+1,it+1⋯sn−1,in−1∈Sn,
we have
[TABLE]
where δJn−1,qu=sq and rq=q if q∈Jn−1;
δJn−1,qu=∂qP(iq≤q) and rq=q−1 if q∈[t,n−1]∖Jn−1;
u(Jn−1)=[srt,it⋯srn−1,in−1]
if srt,it⋯srn−1,in−1 is reduced and
X[1,t−1][δJn−1,tuxt⋯δJn−1,n−1uxn−1]R=0,
u(Jn−1) is undefined otherwise and the summation is over all the Jn−1 such that
u(Jn−1) is defined.
Induction on t.
Suppose t=n−1. If in−1=n, then u=1, Jn−1⊆∅, u(Jn−1)=1. Therefore
[TABLE]
If in−1≤n−1, then u=sn−1,in−1, Jn−1⊆∅, u(Jn−1)=sn−2,in−1. By Lemma 2.2, we have
[TABLE]
for ∂n−1(∂n−2,in−1Sw0n−1)=0.
Suppose t<n−1. If it=t+1, then u=st+1,it+1⋯sn−1,in−1,
{l∈[t,n−2]∣il≤l}={l∈[t+1,n−2]∣il≤l}. So
[TABLE]
If it≤t, let u1=st+1,it+1⋯sn−1,in−1. Then
[TABLE]
where the summation is over all the Jn−1 such that
u(Jn−1) is defined.
Let t=1. Then we have
[TABLE]
By induction hypothesis, we have
[TABLE]
where the summation is over all the Jn−1,…,J1 such that u(Jn−1,…,Jl) is defined for
any 1≤l≤n−1.
Example 2.1**.**
∂uSw06=X[1,5]⋅1≤i<j≤4∑xixj, where u=s1,1s2,1s3,1s4,3.
Proof.
Since we are given ∂uSw06, we should begin with u∈S6. For u=s1,1s2,1s3,1s4,3∈S6, J5⊆{1,2,3,4}. It is easy to see that if J5={1,2,3,4}, then XJ5u=0, so u(J5) is undefined. Let
J5={1,2,3,4}. Then XJ5u=X[1,5],
u(J5)=u=s1,1s2,1s3,1s4,3∈S5, J4⊆{1,2,3}. It is straightforward to see that only when J4=∅ or J4={3} or
J4={2,3}, we have u(J5,J4) is defined. For example, if J4={2}, then XJ4u(J5)=x3, s0,1s2,1s2,1s3,3 is not reduced. If J4=∅, then XJ4u(J5)=1, u(J5,J4)=s1,1s2,1s3,3∈S4, J3⊆{1,2}. In this way, we can list all the possible J5,…,J1 such that u(J5,…,J1) is defined (Table 1).
The result follows immediately from Theorem 1.
Now we begin to construct another formula for Schubert polynomials.
Given n≥2, for any u=s1,i1s2,i2⋯sn−1,in−1∈Sn, for any 1≤r≤n−1, define
[TABLE]
We proceed to define qr,l−1u,Mr,lu by induction on l. Suppose that qr,l−1u,Mr,lu have been defined.
If Mr,lu=∅, then define
[TABLE]
If Mr,lu=∅, then define
[TABLE]
Finally, define
[TABLE]
In other word, qr,ju(j∈[1,mru]) is the largest integer q such that
ir−j∈[iq,q] and q<qr,j−1u.
By definition, we have Qru⊆[1,r], qr,1u>qr,2u>⋯. In particular,
if mru=0, then Qru=∅. For example,
if u=s3,2s5,1s6,4s7,1s8,8s9,5, then
q9,1u=7, q9,2u=5, q9,3u=3, m9u=3.
Lemma 2.3**.**
For any u=s1,i1s2,i2⋯sn−1,in−1,v=s1,j1s2,j2⋯sn−1,jn−1∈Sn, we have
(1)
If ik=jk for any k≤r, then Qtu=Qtv, qt,ju=qt,jv for any 1≤t≤r, 1≤j≤mtu.
2. (2)
If ik=jk for any k≥r, then Qtu∩[r,n−1]=Qtv∩[r,n−1] for any r≤t≤n−1. Moreover,
qt,ju=qt,jv if qt,ju≥r.
3. (3)
If ik=jk for any k=t−1,t and {t−1,t}∩Qru=∅, {t−1,t}∩Qrv=∅ for some 1≤r≤n−1, then
Qru=Qrv, qr,ju=qr,jv for any 1≤j≤mru.
4. (4)
For any j≥1, if ij<j+1, then qj,mjuu>ij−mju−1; If
ij=j+1, then qj,mjuu=ij−mju−1=j.
Proof.
Since (1)-(3) follow immediately from the definition of qt,ju,
we just need to prove (4). If ij=j+1, the claim
is easy. So we may assume that
ij≤j.
If mju=0, then
qj,mjuu=qj,0u=j>j−1≥ij−1=ij−mju−1.
If mju=0, then ij−mju∈[iqj,mjuu,qj,mjuu].
So ij−mju−1<ij−mju≤qj,mjuu.
Given n≥2,
u=s1,i1s2,i2⋯sn−1,in−1∈Sn,
for any
[TABLE]
define
[TABLE]
where [(stmn−1u,…,st2,st1)s1,i1s2,i2⋯sn−2,in−2] is the normal form of the word getting by substituting every sqr,ju,iqr,ju by stjsqr,ju,iqr,ju, 1≤j≤mn−1u.
For each Tn−1 such that u(Tn−1) is defined, define
[TABLE]
where 1≤j≤mn−1u∏x1+qn−1,juP(tj>qn−1,ju)=1 if mn−1u=0.
We proceed to define
u(Tn−1,⋯,Tn−l−1) by induction on l.
Suppose that u(Tn−1,⋯,Tn−l) has been defined. If
u(Tn−1,⋯,Tn−l) is undefined, then Tn−l−1
and
u(Tn−1,⋯,Tn−l−1) are undefined. Otherwise,
u(Tn−1,⋯,Tn−l)∈Sn−l. Say u(Tn−1,⋯,Tn−l)=v=s1,j1⋯sn−l−1,jn−l−1.
For any vector
[TABLE]
we define
u(Tn−1,⋯,Tn−l−1) to be u(Tn−1,⋯,Tn−l)(Tn−l−1)=v(Tn−l−1),XTn−l−1u(Tn−1,⋯,Tn−l)=XTn−l−1v.
Note that the set of Tn−l−1’s depends on u(Tn−1,⋯,Tn−l).
However, for simplicity, we just use the notation Tn−l−1.
In particular, if Qn−1u=∅, then
XTn−1u=X[1,in−1−mn−1u−1].
If tj=qn−1,ju+1 for any 1≤j≤mn−1u, then
u(Tn−1)=s1,i1s2,i2⋯sn−2,in−2∈Sn−1 and
XTn−1u=X[1,in−1−mn−1u−1]x1+qn−1,mn−1uu⋯x1+qn−1,2ux1+qn−1,1u.
For example, we first fix n=11. Let u=s3,2s5,1s6,4s7,1s8,8s9,5∈S11. Then
Q10u=∅, T10=(0), u(T10)=u∈S10,
XT10u=X[1,10], Q9u(T10)={7,5,3},
T9∈{(t3,t2,t1,0)∈N4∣2≤t3≤4,3≤t2≤6,4≤t1≤8}.
If T9=(2,3,4,0), then
u(T10)(T9)=s3,3⋅s5,4s2,1⋅s6,4⋅s7,5s3,1⋅s8,8=s3,1s5,4s6,1s7,5s8,8,
Q8u(T10)(T9)={7,6,5},
T8∈{(t3,t2,t1,0)∈N4∣5≤t3≤6,6≤t2≤7,7≤t1≤8}.
Theorem 2**.**
For any u∈Sn(n≥2), we have
[TABLE]
where the summation is over all the Tn−1,⋯,T1 such that u(Tn−1,…,Tl) is defined for
any 1≤l≤n−1.
Proof.
We first show that for any u=s1,i1⋯sn,in∈Sn+1, we have
∂uSw0n+1=Tn∑XTnu∂u(Tn)Sw0n. Induction on n+1.
If n+1=2, then u=s1,i1, i1=1 or 2. For either case, we have Q1u=∅, so
u(T1)=1 and XT1u=X[1,i1−1]=∂u(x1)=∂uSw02.
If n+1≥3, then
induction on mnu.
If mnu=0, then Qnu=∅, Tn∈{(0)}, u(Tn)=s1,i1s2,i2⋯sn−1,in−1,
XTnu=X[1,in−1].
By applying Lemma 2.2 repeatedly, we have
[TABLE]
If mnu=r≥1, then
u=s1,i1⋯sqn,ru,iqn,rusqn,ru+1,iqn,ru+1⋯sn,in.
Let v=sqn,ru+1,iqn,ru+1⋯sn,in, w=s1,i1⋯sqn,ru−1,iqn,ru−1.
Then by Lemma 2.3, we have mnv=r−1=mnu−1 and qn,jv=qn,ju for any 1≤j≤r−1.
Define
A={(tmnv,…,t1,0)∣in−j≤tj≤qn,jv+1,1≤j≤mnv,tj∈N}={(tmnu−1,…,t1,0)∣in−j≤tj≤qn,ju+1,1≤j≤mnu−1,tj∈N},
B={(tmnu,…,t1,0)∣in−j≤tj≤qn,ju+1,1≤j≤mnu,tj∈N}.
By induction hypothesis, we have
[TABLE]
where the summation is over all the
Tn∈B such that u(Tn) is defined.
By induction hypothesis, we have
[TABLE]
where the summation is over all the Tn−1,⋯,T1 such that u(Tn−1,…,Tl) is defined for
any 1≤l≤n−1.
Corollary 2.1**.**
([14])*
For any w∈Sn, the coefficients of monomials in Sw are nonnegative integers.*
Proof.
Let u=[w−1w0n]. Then Sw=∂uSw0n.
The result follows immediately from Lemma 2.1 and Theorem 1. It also
follows immediately from Theorem 2.
Example 2.2**.**
∂uSw05=1≤i≤j≤3∑xixj, where u=s1,1s2,1s3,2s4,2.
Proof.
For u=s1,1s2,1s3,2s4,2∈S5, by definition, we have q4,1u=2, m4u=1,
T4∈{(1,0),(2,0),(3,0)}.
If T4=(1,0), then
u(T4)=s1,1⋅s1s2,1⋅s3,2=s1,1s2,2s3,2∈S4,
XT4u=1.
If T4=(3,0), then
u(T4)=s1,1⋅s3s2,1⋅s3,2=s1,1s2,1s3,2∈S4,
XT4u=x3.
If T4=(2,0), then
s1,1⋅s2s2,1⋅s3,2=s1s1s3,2 is not reduced,
so u(T4) is undefined.
Let T4=(1,0). Then q3,1u(T4)=1,
m3u(T4)=1,
T3∈{(1,0),(2,0)}. If
T3=(1,0), then u(T4,T3)=s2,2∈S3,
XT3u(T4)=1. In this way, we can list all the possible
T4,…,T1 such that
u(T4,…,T1) is defined (Table 2).
The result follows immediately from Theorem 2.
3 Some combinatorial properties of Schubert polynomials
In this section, we will use Theorem 2 to develop some combinatorial properties of Schubert polynomials.
For any u=s1,i1⋯sn−1,in−1∈Sn, j∈[1,n−1], define
[TABLE]
In particular, if Qju=∅, then mju=0 and
Xju=X[1,ij−1].
For any commutative word W=x1k1⋯xn−1kn−1 (each ki∈N),
define degxt(W)=kt. It is clear that for any j,t,p∈[1,n−1],p<t, we have degxt(Xju)≤1 and degxt(Xpu)=0.
Lemma 3.1**.**
Let u=s1,i1⋯sn−1,in−1∈Sn, v=s1,j1⋯sn−1,jn−1∈Sn, r∈[1,n−1].
If ik=jk for any k≥r, then
(1)
degxk(Xtu)=degxk(Xtv)* for any t∈[r+1,n−1], k∈[r+1,t].*
2. (2)
If we have also ir−1≤jr−1, then degxr(Xtu)≥degxr(Xtv) for any t≥r.
Proof.
To prove (1), we only need to show that degxk(Xtu)≤degxk(Xtv) for any t∈[r+1,n−1], k∈[r+1,t].
Note that Xtu=X[1,it−mtu−1]x1+qt,mtuux1+qt,mtu−1u⋯x1+qt,1u.
If degxk(Xtu)=0, we are done. If it=t+1, then degxk(Xtu)=degxk(Xtv).
So we may assume that degxk(Xtu)=1, it≤t.
If degxk(X[it−mtu−1])=1, then it−mtu−1≥k≥r+1. By definition, we have
it−mtu∈[iqt,mtuu,qt,mtuu], so qt,mtuu≥it−mtu≥r+2, Qtu⊆[r+2,n]⊆[r,n].
By Lemma 2.3, we have qt,pu=qt,pv for any 1≤p≤mtu. So mtv≥mtu. Moreover, if mtv>mtu,
then it−mtu−1∈[iqt,mtu+1v,qt,mtu+1v], qt,mtu+1v≥it−mtu−1≥r+1 but
qt,mtu+1v∈/Qtu,
which contradicts with Lemma
2.3. Therefore Qtu=Qtv, Xtu=Xtv, degxk(Xtu)≤degxk(Xtv).
If xk=xqt,lu+1 for some l∈[1,mtu], then
qt,lu=k−1≥r+1−1≥r. By Lemma 2.3, we
have qt,lu∈Qtu∩[r,n−1]=Qtv∩[r,n−1] and
qt,lu=qt,lv. Hence xqt,lv+1=xqt,lu+1=xk,
degxk(Xtu)≤degxk(Xtv).
To prove (2), we only need to show that if for some t≥r, degxr(Xtv)=1, then degxr(Xtu)=1.
If it=t+1, then we are done. So we may assume that it≤t.
If degxr(X[jt−mtv−1])=1, then jt−mtv−1≥r. By definition, we have
jt−mtv∈[jqt,mtvv,qt,mtvv], so qt,mtvv≥jt−mtv≥r+1, Qtv⊆[r+1,n]⊆[r,n].
By Lemma 2.3, we have Qtu∩[r,n]=Qtv∩[r,n]=Qtv. Moreover,
it−mtv−1=jt−mtv−1≥r and thus for any q<r, it−mtv−1∈/[iq,q].
So Qtu=Qtv, Xtu=Xtv, degxr(Xtu)=1.
If xr=xqt,pv+1 for some p∈[1,mtv], then qt,pv=r−1 and qt,lv≥r for any l≤p−1.
Moreover, it−p=jt−p∈[jqt,pv,qt,pv]=[jr−1,r−1]⊆[ir−1,r−1] and
it−p=jt−p∈/[jl,l]=[il,l] for any l∈[r,qt,p−1v−1]=[r,qt,p−1u−1]. So qt,pu=r−1, xr=xqt,pu+1,
degxr(Xtu)=1.
Let X={x1,…,xn−1}. Define an order < on the free commutative monoid [X] as follows:
For any U=x1k1⋯xn−1kn−1∈[X], V=x1l1⋯xn−1ln−1∈[X],
[TABLE]
For any f∈Z[x1,…,xn−1], define fˉ to be the leading monomial of f with respect to the order <. If the coefficient of fˉ=1, then we say that f is monic. For example,
if f=3x32+2x3x7−7x5x7, then fˉ=x5x7.
Lemma 3.2**.**
For any u=s1,i1⋯sn−1,in−1∈Sn(n≥2), we
have ∂uSw0n is monic and ∂uSw0n=Xn−1u⋯X1u,
where Xju=X[1,ij−mju−1]x1+qj,mjuux1+qj,mju−1u⋯x1+qj,1u for any j∈[1,n−1].
Proof.
Induction on n. If n=2, then ∂uSw0n=X[1,i1−1]=X1u.
Suppose the lemma holds for any k≤n. Let k=n+1,
u=s1,i1⋯sn,in∈Sn+1, u1=s1,i1⋯sn−1,in−1∈Sn.
If mnu=0, then by the proof of Theorem 2, we have
[TABLE]
By induction hypothesis, we have
[TABLE]
If mnu>0,
then let A={(tmnu,…,t1,0)∣in−j≤tj≤qn,ju+1,1≤j≤mnu,tj∈N}.
We have
[TABLE]
So we just need to show that if
Tn=(tmnu,⋯,t1,0)=(qn,mnuu+1,…,qn,1u+1,0), then
x1+qn,mnuuP(tmnu>qn,mnuu)⋯x1+qn,1uP(t1>qn,1u)X[1,in−mnu−1]∂u(Tn)(Sw0n)<Xnu⋯X1u if u(Tn) is defined.
Let W(Tn)=XTnuXn−1u(Tn)⋯X1u(Tn)=x1+qn,mnuuP(tmnu>qn,mnuu)⋯x1+qn,1uP(t1>qn,1u)X[1,in−mnu−1]⋅Xn−1u(Tn)⋯X1u(Tn).
Suppose that tl=qn,lu+1 for any 1≤l<r (r≥1) and tr∈[iqn,ru,qn,ru].
Then by the definition of u(Tn), we have
[TABLE]
By
using the
Gröbner-Shirshov basis of Sn, we have jt=it for any t≥qn,ru+1
and iqn,ru<tr+1=jqn,ru. By Lemma 3.1,
we have
degx1+qn,ru(Xju)≥degx1+qn,ru(Xju(Tn)) for
any n−1≥j≥qn,ru+1
and
degxt(Xju)=degxt(Xju(Tn)) for any n−1≥j≥t≥qn,ru+2.
Since degxqn,ru+1(XTnu)=0<1=degxqn,ru+1(Xnu),
we have
degxt(XTnu⋅Xn−1u(Tn)⋯X1u(Tn))=degxt(Xnu⋯X1u)
for any t≥qn,ru+2 and
degxqn,ru+1(XTnu⋅Xn−1u(Tn)⋯X1u(Tn))<degxqn,ru+1(Xnu⋯X1u).
Since ∂uSw0n is
homogeneous and the coefficients of ∂uSw0n in Theorem 2 are nonnegative, the lemma
follows.
Lemma 3.3**.**
For any u=s1,i1⋯sn−1,in−1∈Sn, the following statements are equivalent:
(i)
stu* is reduced.*
2. (ii)
it−1<it.
3. (iii)
degxt(∂uSw0n)>degxt+1(∂uSw0n).
Moreover, if it−1<it, then degxtXtu=1 and degxtXju≥degxt+1Xju for any j∈[t+1,n−1],
if it−1≥it, then degxtXtu=0 and degxtXju≤degxt+1Xju for any j∈[t+1,n−1].
Proof.
(i)⇒ (ii) By the Gröbner-Shirshov basis of Sn, we have
stu=s1,i1⋯st−2,it−2⋅st⋅st−1,it−1st,it⋅st+1,it+1⋯sn,in.
Suppose that it−1≥it.
If it−1=t≥it, then
[TABLE]
So stu is not reduced.
If t−1≥it−1≥it, then
[TABLE]
So stu is not reduced.
Consequently, if stu is reduced, then it−1<it.
(ii) ⇒ (i) If it−1<it, then by similar reasoning as above, we have
stu=s1,i1⋯st−2,it−2⋅st−1,it−1⋅stst−1,it−1⋅st+1,it+1⋯sn,in∈Sn,
so stu is reduced.
(ii) ⇒ (iii) Note that Xju=X[1,ij−mju−1]x1+qj,mjuux1+qj,mju−1u⋯x1+qj,1u.
Since ∂uSw0n=Xn−1u⋯X1u, it is enough to
show that degxtXju≥degxt+1Xju for any j∈[t+1,n−1] and
degxtXtu=1(>0=degxt+1Xtu).
If degxt+1(X[1,ij−mju−1])=1 for some j≥t+1, then degxt(X[1,ij−mju−1])=1.
If xt+1=x1+qj,lu for some l∈[1,mju], then qj,lu=t,
ij−l∈[it,t],
so ij−l−1∈[it−1,t−1]⊆[it−1,t−1].
By definition, qj,l+1u=t−1, and thus x1+qj,l+1u=xt.
By the above reasoning, we have degxtXju≥degxt+1Xju for any j∈[t+1,n−1].
If it=t+1, then Qtu=∅ and Xtu=X[1,it−1]=X[1,t]. So degxtXtu=1.
If it≤t, then it−1∈[it−1,t−1]. By definition, we have
qt,1u=t−1, xt=x1+qt,1u. So degxtXtu=1.
(iii) ⇒ (ii) We just need to show that if it−1≥it, then
degxt(∂uSw0n)≤degxt+1(∂uSw0n).
Since ∂uSw0n=Xn−1u⋯X1u, it is enough to
show that degxtXju≤degxt+1Xju for any j∈[t+1,n−1] and
degxtXtu=0(=degxt+1Xtu).
If degxt(X[1,ij−mju−1])=1 and degxt+1(X[1,ij−mju−1])=0 for some j≥t+1.
Then t=ij−mju−1. Since it−1≥it, we have it≤it−1≤t.
If mju=0, then for any r∈[1,j−1], t=ij−mju−1=ij−1∈/[ir,r], which contradicts with t∈[it,t]. If
mju>0, then ij−mju=t+1∈[iqj,mjuu,qj,mjuu],
qj,mjuu≥t+1. Moreover, by the definition of mju, we know that
t=ij−mju−1∈/[ir,r] for any r∈[1,qj,mjuu−1], which contradicts with t∈[it,t]. So
if degxt(X[1,ij−mju−1])=1, then degxt+1(X[1,ij−mju−1])=1.
If xt=xqj,lu+1 for some l∈[1,mju], then
ij−l∈[it−1,t−1]⊆[it,t] and
ij−l∈/[iq,q] for any q∈[t,qj,l−1u−1]. This is possible
only if qj,l−1u=t, which means that xt+1=xqj,l−1u+1.
By the above reasoning, we have degxtXju≤degxt+1Xju for any j∈[t+1,n−1].
Since it≤it−1≤t, we have
it−1>it−1 and it−1≤t−1. By the definition of qt,1u and Xtu, we have t−1∈/Qtu and
degxt(Xtu)=0.
Lemma 3.4**.**
For any v=s1,i1⋯sn−1,in−1∈Sn, if degxt(∂vSw0n)>degxt+1(∂vSw0n),
then
∂stvSw0n=∂st(∂vSw0n).
Proof.
If degxt(∂vSw0n)>degxt+1(∂vSw0n),
then by Lemma 3.3, we have it−1<it.
Say ∂vSw0n=X1v⋯Xn−1v. Let
[TABLE]
Then by Lemma 3.3, we have t∈T and degxtXjv=degxt+1Xjv for any j∈[1,n−1]∖T. We may assume
that T={pk∣1≤k≤l}⊆[t,n−1], p1=t. Then
[TABLE]
where q1,…,qn−l∈[1,n−1]∖T.
On the other hand,
let u=stv. By Lemma 3.3, u is a reduced word and
u=stv=s1,i1⋯st−2,it−2⋅st−1,it−1⋅stst−1,it−1⋅st+1,it+1⋯sn,in=s1,j1⋯sn−1,jn−1∈Sn, jt=it−1, jt−1=it−1≤t, jp=ip
for any p∈[1,n−1]∖{t−1,t}.
Say ∂uSw0n=X1u⋯Xn−1u.
By the definition of Xju and Lemma 2.3, we have Xju=Xjv for any j∈[1,t−2].
The proof will proceed in steps.
(i) Xtu=Xt−1v,Xt−1u=xtXtv.
Since u=s1,i1⋯st−2,it−2⋅st−1,it−1⋅stst−1,it−1⋅st+1,it+1⋯sn,in,v=s1,i1⋯sn−1,in−1 and it−1<it, it is straightforward to show that
Qtv=Qt−1u∪{t−1}, Qt−1v=Qtu, qt,1v=t−1. Therefore
Xtu=X[1,jt−mtu−1]x1+qt,mtuu⋯x1+qt,1u=X[1,it−1−mt−1v−1]x1+qt−1,mt−1vv⋯x1+qt−1,1v=Xt−1v and
Xt−1u=X[1,jt−1−mt−1u−1]x1+qt−1,mt−1uu⋯x1+qt−1,1u=X[1,(it−1)−(mtv−1)−1]x1+qt,mtvv⋯x1+qt,2v=xtXtv.
(ii) For any j∈[t+1,n−1], if j∈T∖{t}, then we have
Xju=xtXjv⋅xt+1.
If for some j∈[t+1,n−1], j∈T∖{t},
ij=j+1, then Xjv=X[1,j], which contradicts with j∈T. So we have ij≤j. By Lemma 2.3, we have qj,mjvv>ij−mjv−1.
Note that
Xjv=X[1,ij−mjv−1]x1+qj,mjvv⋯x1+qj,1v. There are
two cases:
Case 1. If xt=xij−mjv−1, then t=ij−mjv−1∈/[ip,p] for any p∈[1,qj,mjvv−1] and qj,mjvv−1≥ij−mjv−1=t. In particular, t∈/[it,t], so it=t+1, it−1≤it−1=t.
We have
ij−(mjv+1)=t∈[it−1,t]=[jt,t], ij−(mjv+1)∈/[ip,p] for any p∈[t+1,qj,mjvv−1].
Therefore qj,mjv+1u=t, qj,ku=qj,kv for any k∈[1,mjv]. Moreover, ij−(mjv+1)−1=t−1∈/[t,t−1]=[it−1,t−1] and t−1∈/[ip,p] for any p≤t−2. It follows that Qju=Qjv∪{t} and Xju=X[1,ij−(mjv+1)−1]⋅x1+qj,mjuu⋅x1+qj,mjvv⋯x1+qj,1v=X[1,ij−mjv−2]⋅xt+1⋅x1+qj,mjvv⋯x1+qj,1v=xtXjv⋅xt+1.
Case 2. If xt=x1+qj,kv for some k∈[1,mjv], then
qj,kv=t−1. Since j∈T, we have qj,k−1v>t, ij−k∈[it−1,t−1]
and ij−k∈/∪t≤p≤qj,k−1v−1[ip,p].
Thus ij−k∈[it−1,t]=[jt,t]
and ij−k∈/∪t+1≤p≤qj,k−1v−1[jp,p].
By definition, we have qj,pu=qj,pv for any p∈[1,k−1] and
qj,ku=t. Moreover, since ij−k∈/[it,t], we have
ij−k−1∈/[it−1,t−1]=[jt−1,t−1]. So t−1∈/Qju.
It follows immediately that qj,lv=qj,lu for any l∈[k+1,mjv] and
mjv=mju.
Therefore Qju∖Qjv={t}, Qjv∖Qju={t−1},
Xju=xtXjv⋅xt+1.
(iii) For any j∈[t+1,n−1]∖T, we have Xju=Xjv.
For any j∈[t+1,n−1]∖T, we have
degxt(Xjv)=degxt+1(Xjv).
If for some j∈[t+1,n−1]∖T,
ij=j+1, then Xju=Xjv=X[1,j].
So we may assume that ij≤j. By Lemma 2.3, we have qj,mjvv>ij−mjv−1.
There are two cases:
Case 1. If degxt(Xjv)=degxt+1(Xjv)=0, then
ij−mjv−1≤t−1. Moreover, if qj,mjvv≥t+1, then
since ij−mjv−1∈/[ip,p] for any p∈[1,qj,mjvv−1] by the definition of mjv, we have
ij−mjv−1∈/[it−1,t−1]∪{t}.
Since it−1≤it−1, we have
ij−mjv−1∈/∪1≤p≤t−2[ip,p]∪[it−1,t−1]∪[it−1,t]∪∪t+1≤p≤qj,mjvv−1[ip,p],
so mju=mjv, Qju=Qjv, Xju=Xjv.
If qj,mjvv≤t−2, then there is some k∈[0,mjv−1] such that
qj,k+1v≤t−2, qj,kv≥t+1.
Then ij−k−1∈/∪qj,k+1v+1≤p≤qj,kv−1[ip,p] and ij−k−1∈[iqj,k+1v,qj,k+1v].
In particular, ij−k−1∈/[it−1,t−1]∪{t}=[it−1,t]∪[it−1,t−1]. So {t−1,t}∩Qju=∅. By Lemma 2.3, we have
mju=mjv, Qju=Qjv, Xju=Xjv.
Case 2. degxt(Xjv)=degxt+1(Xjv)=1.
If ij−mjv−1≥t+1, then qj,mjvv>ij−mjv−1≥t+1.
By similar reasoning as case 1, we have mju=mjv, Qju=Qjv, Xju=Xjv.
If ij−mjv−1=t, then qj,mjvv>ij−mjv−1=t, qj,mjvv+1>t+1, which
contradicts with degxt+1(Xjv)=1.
If ij−mjv−1<t, then there is some k∈[1,mjv−1] such that 1+qj,kv=t+1, 1+qj,k+1v=t. So
ij−k∈[it,t], ij−k−1∈[it−1,t−1]. Consequently,
ij−k∈[it−1,t]=[jt,t], ij−k−1∈[it−1,t−1]=[jt−1,t−1]. By similar reasoning as case 1, we have mju=mju, Qju=Qjv, Xju=Xjv.
Corollary 3.1**.**
For any reduced word u=st1⋯stp∈Sn, we have
[TABLE]
Lemma 3.5**.**
For any u,v∈Sn, if u=v, then we have
∂uSw0n=∂vSw0n.
Proof.
If ∣u∣=∣v∣, then since ∂uSw0n is a
homogeneous polynomial of degree 21n(n−1)−∣u∣, we have
∂uSw0n=∂vSw0n. Assume ∣u∣=∣v∣.
Induction on ∣u∣. If ∣u∣=1, it is trivial.
Suppose
∣u∣=∣v∣≥2, u=stu1∈S∗, v=skv1∈S∗,
u,v,u1,v1 are all normal forms in Sn and
∂u1Sw0n=x1l1⋯xn−1ln−1xnln,lt>lt+1,ln=0,∂v1Sw0n=x1p1⋯xn−1pn−1xnpn,pk>pk+1,pn=0.
By Lemma 3.4, we have
[TABLE]
[TABLE]
If t=k, then u1=v1, by induction hypothesis, there exists some
q∈[1,n−1] such that lq=pq. It follows immediately that
∂uSw0n=∂vSw0n.
If t=k, then we may assume that t<k.
Since u,v,u1,v1 are all in normal form, we have u=st,it⋯sn−1,in−1, u1=st−1,itst+1,it+1⋯sn−1,in−1,
v=sk,jk⋯sn−1,jn−1. Moreover,
stv=stsk,ik⋯sn−1,in−1
is a normal form, hence stv
is reduced. By Lemmas 3.3 and 3.4, we have
degxt(∂uSw0n)=lt+1≤lt−1=degxt+1(∂uSw0n)
and
degxt(∂vSw0n)>degxt+1(∂vSw0n).
Consequently, ∂uSw0n=∂vSw0n.
Remind that Bx:={x1k1⋯xn−1kn−1∣ki+i≤n,1≤i≤n−1}.
Define the map φ to be φ:Sn⟶Bx, φ(u)=∂uSw0n.
By Lemma 3.5, we know that φ is an injective map. But the cardinal of Bx is n!, so
φ is a bijection. The inverse of φ can be easily
constructed by Lemmas 4.2 and 4.3.
For any u∈Sn(n≥2), t∈[1,n−1], we have the following combinatorial properties
of Schubert polynomials:
(i)
∂uSw0n* is monic and ∂uSw0n=Xn−1u⋯X1u,
where Xju=X[1,ij−mju−1]1≤k≤mju∏x1+qj,ku for any j∈[1,n−1].
Moreover, the map φ:Sn⟶Bx, φ(u)=Xn−1u⋯X1u is a
bijection. In particular, for any v∈Sn, if u=v, then
∂uSw0n=∂vSw0n.*
2. (ii)
stu* is reduced if and only if degxt(∂uSw0n)>degxt+1(∂uSw0n).
Moreover, if degxt(∂uSw0n)>degxt+1(∂uSw0n),
or equivalently, stu is reduced,
then
∂stuSw0n=∂st(∂uSw0n).*
For any u∈Sn, define
[TABLE]
Then we have
Corollary 3.2**.**
For any W∈Bx, if degxt(W)>degxt+1(W), then ∂tSW=S∂tW.
Proof.
Since W∈Bx, by Theorem 3, there is a u∈Sn such that W=∂uSw0n.
If degxt(W)>degxt+1(W), then stu is reduced and
∂tSW=∂t∂uSw0n=∂t∂uSw0n=∂tW.
By the definition of SW, we are done.
Since
∂uSw0n is monic, we easily get the following corollaries.
Corollary 3.3**.**
([15])*
The Schubert polynomials ∂uSw0n, as u varies over all
permutations in Sn, form an additive basis of the free Z-module ⊕b∈BxZb.*
Corollary 3.4**.**
([15])*
The Schubert polynomials Su, as u varies over all
permutations in S∞, form an additive basis of the free polynomial ring
Z[x1,x2,⋯,xn,⋯].*
4 Algorithms for multiplication of Schubert polynomials
For any Schubert polynomials Su, Sv,
by Corollary 3.4, we know that there are structure constants cu,vw∈Z such that
[TABLE]
It is well known that the coefficients are all nonnegative (for example, see [9]),
but there is no combinatorial proof yet.
One of the most famous formula for multiplications of Schubert polynomials is Monk’s formula [16, 9]:
[TABLE]
where the summation is over all v such that v=w⋅spsp+1⋯sq−2sq−1,p and
l(v)=l(w)+1, where p≤k and q>k. For example, Ss2⋅Ss2=Ss1s2+Ss3s2.
We will offer algorithms to calculate the structure constants in the sequel. However, for simplicity of the algorithms, we will use the notation Su. Since Ssk=i∈[1,k]∑xi, we have Ssk=Sxk. Assume that u∈Sn−k. Then by Monk’s formula, we have
[TABLE]
where
the summation is over all v such that w0nv−1=w0nu−1⋅spsp+1⋯sq−2sq−1,p,
l(w0nv−1)=l(w0nu−1)+1 where p≤k and q>k.
Let W=x1j1⋯xn−1jn−1∈Bx, jn=0.
If for some p1,⋯,pm∈[1,n−1], p1<p2<⋯<pm, jpt=jpt+1+1 for any t∈[1,m], then
∂pm⋯∂p2∂p1SW=SV,
where V=xp1⋯xpmW.
Proof.
Since W∈Bx, by Theorem 3, there is a u∈Sn such that W=∂uSw0n.
Induction on m. If m=1, then
degxp1(∂uSw0n)=jp1>jp1+1=degxp1+1(∂uSw0n).
So ∂p1SW=∂p1∂uSw0n=∂p1∂uSw0n=∂p1W=V.
By the definition of SV, we have ∂p1SW=SV.
Suppose the assertion holds for any k<m. Then ∂pm−1⋯∂p2∂p1SW=SV1, ∂pmSV1=SV,
where V1=xp1⋯xpm−1W and
V=xpmV1=xp1⋯xpmW.
For any W=x1j1⋯xn−1jn−1∈Bx, j1≥j2≥⋯≥jn−1, define
[TABLE]
where pk,i<pk,j if i<j. In particular, P0W=[1,n−1].
If PkW=∅ for some k>0, then define
[TABLE]
Then we have the following lemma:
Lemma 4.2**.**
For any W=x1j1⋯xn−1jn−1∈Bx, j1≥j2≥⋯≥jn−1, let
m=max{k∈[0,n−1]∣PkW=∅}. If m>0, then for any k∈[1,m], we have
[TABLE]
In particular, ∂vmWvm−1W⋯v1WSw0n=W.
Proof.
Let Vk=xpk,tk⋯xpk,1⋯xp2,t2⋯xp2,1xp1,t1⋯xp1,1x1n−1x2n−2⋯xn−11.
Induction on k. If k=1, then by Lemma 4.1,
we have SV1=∂p1,t1⋯∂p1,1Sx1n−1x2n−2⋯xn−11=∂p1,t1⋯∂p1,1(x1n−1x2n−2⋯xn−11)=V1.
Suppose the lemma holds for any q<k, k≥2.
If l,l+1∈PkW, then l,l+1∈Pk−1W.
If l∈PkW, l+1∈/PkW, then there is some integer t such that
l+1∈PtW, l+1∈/Pt+1W. It is clear that t≤k−1.
Moreover, by the definition of PkW, we have n−l−jl≥k, n−(l+1)−jl+1=t. So
n−(l+1)−t=jl+1≤jl≤n−l−k. It follows that k−1≥t≥k−1, i.e., t=k−1.
Therefore l,l+1∈Pk−1W and
degxl(∂vk−1W⋯v1WSw0n)=n−l−(k−1)=n−(l+1)−(k−1)+1=degxl+1(∂vk−1W⋯v1WSw0n)+1.
By Lemma 4.1 and induction hypothesis, the lemma follows.
Lemma 4.3**.**
Given t∈[0,n−2], Wt=x1j1⋯xn−1jn−1∈Bx such that
j1+1≥j2+2≥⋯≥jt+t≥max{jk+k∣k∈[t+1,n−1]}, define
[TABLE]
where k∈[t+1,n−1] is as small as possible such that jk+k=max{jk+k∣k∈[t+1,n−1]}. Then
we have ∂vt+1WtSWt+1=Wt.
In particular, for any W0∈Bx, we can construct
v1W0,W1,⋯,vn−2Wn−3,Wn−2 such that ∂viWi−1SWi=Wi−1, which implies that
∂v1W0⋯∂vn−3Wn−4∂vn−2Wn−3SWn−2=W0. Moreover, if
Wi=x1li,1⋯xn−1li,n−1, then li,1+1≥li,2+2≥⋯≥li,i+i≥max{li,k+k∣k∈[i+1,n−1]}.
Proof.
Induction on k−t.
If k=t+1, then Wt+1=Wt, vt+1Wt=sk−1,t+1=1, ∂vt+1WtSWt+1=Wt.
If k>t+1. Let W′=x1j1⋯xk−2jk−2⋅xk−1jk+1xkjk−1⋅xk+1jk+1⋯xn−1jn−1∈Bx. Since jk+k>jk−1+k−1, we have jk+1>jk−1. By Corollary 3.2, we have ∂k−1SW′=Wt. By induction hypothesis, we have ∂sk−2,t+1SWt+1=W′, i.e., ∂sk−2,t+1SWt+1=SW′. Consequently,
∂sk−1,t+1SWt+1=∂sk−1SW′=Wt. The lemma follows immediately.
Proof.Sx3x4=∂3Sx32x4=∂3,2Sx23x4=∂3,1Sx14x4=∂3,1∂3Sx14x32=∂3,1∂3,2Sx14x23=∂3,1∂3,2∂3∂4,3Sw05=∂uSw05,
where u=s3,1s3,2s3s4,3. By the Gröbner-Shirshov basis of S6, we have [u]=s1,1s2,1s3,1s4,3.
In fact, Lemmas 4.2, 4.3 together offer an algorithm to
construct a reduced word u∈Sn such that ∂uSw0n equals
an arbitrary commutative word in Bx. And the
Gröbner-Shirshov basis of Sn offers an algorithm to rewrite any word u∈Sn to its
normal form, which can be easily applied to calculate the Schubert Polynomial by Theorem 1 or Theorem 2.
Moreover, Lemma 3.2 offers an algorithm to write down the leading monomial
of the Schubert polynomial ∂uSw0n for any u∈Sn.
For any W∈Bx, f∈Z[x1,⋯xn], denote by cW(f) the coefficient of W in f.
Recall that for any u∈Sn,
[TABLE]
Now we can offer an algorithm to calculate the structure constants cu,vw as follows:
Algorithm 1. For any u,v∈S∞, by Lemma 3.2, we have
SuSv=uv and SuSv is monic. By Lemma 4.3 and then Lemma 4.2, we can find a w1∈S∞, such that w1=uv. Say w1∈Sn. Then using the Gröbner-Shirshov basis of Sn, we can write down the normal form of w1.
By Theorem 2 (or Theorem 1), we calculate
Sw1=∂w1Sw0n.
Then we construct an w2 such that w2=SuSv−Sw1<uv.
Since each Sw for any w∈S∞ is monic and the leading monomials decrease in each step, the algorithm works.
In particular,
For any u,v∈S∞, we have
[TABLE]
where
cu,vw=cw(SuSv)−z∈{z∣w<z≤uv}∑cu,vzcw(Sz)
if
w<uv.
By applying Monk’s formula and Lemma 3.2, we also have another algorithm to calculate the structure constants cu,vw as follows:
Algorithm 2. Induction on min{∣u∣,∣v∣}. If min{∣u∣,∣v∣}≤1, then by Monk’s formula, we are done.
If min{∣u∣,∣v∣}=t≥2,
say ∣u∣=t.
Induction on u. If u=x1t, then Su=x1t=Sx1⋅Sx1t−1. By induction hypothesis, we have formula
[TABLE]
If u>x1t, then degxku≥1 for some k (for example,
choose the smallest one).
By Monk’s formula, we have
[TABLE]
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