Fomin- Kirillov Algebra Sirous Homayouni York University - - PowerPoint PPT Presentation

Fomin- Kirillov Algebra

Sirous Homayouni

York University shomayou@mathstat.yorku.ca

July 17, 2015

Overview

Fomin-Kirillov Algebra Definition Grobner Basis Monomial Ordering Buchberger Algorithm Lexicographic Ordering S-Polynomial Reduction of S-Polynomial Lemma Degree 3 elements of GB Proposition Degree≥ 3

Definition

The Fomin-Kirilov Algebra FK(n) is a non-commutative quadratic algebra over a field with generators xij = −xji for 1 ≤ i < j ≤ n with the following relations: i : x2

ij

for i < j; ii : xijxkl − xklxij = 0 whenever {i, j} ∩ {k, l} = φ, i < j, and k < l; iii : xijxjk − xjkxik − xikxij = 0, iii’ : xijxik − xjkxij + xikxjk = 0 for i < j < k. (1)

In other words, Let V be the vector space spanned by the generators xij. Then FK(n) is a quotient of the tensor algebra T(V ) = ⊕n≥0V ⊗n, the free associative algebra on the generators

• f FK(n), where the quotient is over the homogeneous ideal

generated by the relations in the FK(n) definition. While different sets of polynomials can generate the ideal, among them however, there is a set with specific properties call Grobner basis (GB) defined as follows.

Definition

A Grobner basis for an ideal I in the polynomial ring F[x1, · · · , xn] is a finite set of generators {g1, · · · , gm} for I whose leading terms generate the ideal of all leading terms in I, i.e., I = (g1, · · · , gm) and LT(I) = (LT(g1), · · · , LT(gm)).

From the phrase ”leading term” in Definition 2 it is understood that GB depends on the monomial ordering we choose. Therefore we need to define monomial ordering.

Definition

A monomial ordering is a well ordering ”≥” on the set of monomials that satisfies mm1 ≥ mm2 for monomials m, m1, m2. Equivalently, a monomial ordering may be specified by defining a well ordering on the n-tuple α = (a1, · · · , an) ∈ Z n of multi-degrees

• f monomials Axa1

1 · · · xan n that satisfies α + γ ≥ β + γ if α ≥ β.

As mentioned earlier different monomial orderings result in different Grobner bases. One of the monomial orderings is Lexicographic ordering.

Definition

The Lexicographic ordering of polynomials that we use is defined by first introducing a variable ordering by

• 1. if the 2nd index of the 1st variable of a monomial M1 is

smaller than that of monomial M2, then M1 comes first, M1 < M2,

• 2. if the 2nd indexes of the 1st variables of M1 and M2 are

equal, then look at the 1st index, the one with bigger 1st index comes first. Then with the above variable ordering, the following rule completes the definition of our lexicographic monomial ordering (lex),

◮ If the 1st variables of M1 and M2 happens to be of the same

• rder, then look at the next variable.

Example

x23x13x23 < x14x23x13, x23x13x23 < x13x23x13, x23x24 < x23x14.

Now with our lexicographic ordering the monomial that comes first is called ”leading monomial” (LM), and the coefficient of the leading monomial is called leading coefficient (Lc). Now, we can use Buchberger’s algorithm to calculate Grobner basis. However before introducing Buchberger’s algorithm we need to define S − polynomial.

Definition

Let f , g ∈ k[x1, · · · , xn] be nonzero polynomials.

• 1. If multideg (f ) = α and multideg (g) = β, then let

γ = (γ1, · · · , γn), where, γi = max {αi, βi} for each i. We call xγ the ”least common multiple” of LM(f ) and LM(g), written xγ = LCM(LM(f ), LM(g)).

• 2. The S − polynomial of f and g is the combination

S(f , g) =

xγ LM(f )f − xγ LM(g)g.

Definition

”Reduction of S-polynomial S(f , g) with respect to the list of generators G = [g1, g2, · · · , gt]”, denoted S(f , g)

G is defined as

the remainder of division of S(f , g) by the ordered elements of G.

Buchberger’s Algorithm

Buchberger’s algorithm is a method of generating the elements of GB and works on the basis of a theorem according to which if the ideal I =< f1, · · · , fs >= {0} is a polynomial ideal, then a GB for I can be constructed in a finite number of steps by the following algorithm.

Input: F = (f1, · · · , fs) Output: a Grobner basis G = (g1, · · · , gt) for I, with F ⊂ G G := F REPEAT G ′ : G FOR each pair {p, q}, p = q in G ′ DO S := S(p, q)

G ′

IF S = 0 THEN G := G ∪ S UNTIL G = G ′ In the above, S(p, q)

G ′

is the remainder of division of S−polynomial S(p, q) by the ordered elements of G ′. We call S(p, q)

G ′

”reduction of S(p,q) with respect to the list G ′”.

With our Lexicographic ordering we show that elements of any degree d, of a GB generated by Buchberger’s algorithm for the ideal I, have a special graphical property that we call it ”star” defined below.

Definition

A monomial is called z-star monomial if for a fix z, all its variables are of the form xαz, where 1 ≤ α < z. A homogenous polynomial where all monomials are z-star with the same z, is called a z-star polynomial.

Example

x23x13x23 + x13x23x13 is a 3-star polynomial, while x23x12x23 + x13x23x13 is not a z-star polynomial.

Theorem

Let T (d) be the set of degree d elements of a GB for FK(n). Then, for d ≥ 3 , T (d) contains only z-star elements.

Lemma

Let M be a z-star monomial, and xij a generator of FK(n) such that i, j = z. Then, under the reduction algorithm, xijM, is splitted into one or more z-star monomials (with the same z) and a non-star monomial, i.e., xijM → a non-star monomial + a z-star

• polynomial. Moreover
• 1. The non-star monomial is equal to [τijM]xij, where τij is a

transposition in the group of permutations.

• 2. The z-star polynomial is

k(M)ij k for k = 1, 2, 3, · · · , where

(M)ij

k is made of M as following.

Let M = xa1zxa2z · · · xakz · · · xadz. For k = 1, · · · , d, if ak = i, j then (M)ij

k = 0; otherwise

if ak = i, then (M)ij

k = −

• τij(xa1z · · · xak−1z)
• xizxjz(

M xa1z···xak z );

if ak = j, then (M)ij

k =

• τij(xa1z · · · xak−1z)
• xjzxiz(

M xa1z···xak z )

The set of degree 3 elements of a GB for FK(n) is T (3) =

• g(3) = xbzxazxbz + xazxbzxaz : 1 ≤ a < b < z ≤ n
• ,

i.e., T (3) contains only z-star elements for different values of z ≤ n. This Proposition is the base of induction and in the next chapter the inductive step will be developed.

The overlap between relations (ii) and (i) results in zero as follows, xabxcz − xczxab with xczxcz yields P = xczxabxcz that under the algorithm it reduces to zero. Overlap between relations (ii) and (iii) results in zero as follows xαβxab − xabxαβ for 1 ≤ α < β < b < z ≤ n and a < b with xabxbz − xbzxaz − xazxab gives P = −xαβxbzxaz − xαβxazxab + xabxαβxbz which goes zero under reduction algorithm. Also overlap of xabxbz − xbzxaz − xazxab with xbzxcz′ − xcz′xbz for z < z′ results in P = −xabxcz′xbz + xbzxazxcz′ + xazxabxcz′ which under reduction goes zero. The only possibility for getting non-zero result is for overlap between relations (i) and (iii).

Remark: For elements of Grobner basis, p and q, the calculation of S(p, q)

G ′

is in general is complicated and involves calculation of a lot of cases for p and q. However when we restrict our self to special case when q is a z-star element, and when we are looking for a probable non-star out come for our calculation, then (as is shown below) every thing is reduced to only three cases for p, i.e, three relations (ii), (iii) and (iii′) . Therefore, in case of z-star q, we need to check only S(xijxαz − xαzxij, q), S(xiαxαz − xαzxiz − xizxiα, q), and S(xiαxiz − xαzxiα + xizxαz, q) where 1 ≤ i < j < α < z ≤ n, for a non-star out come.

The reason of the above is explained in the following items:

• 1. Since S(p, q) for z-star p, q can’t be non-star, we should check

S(p, q) for only z-star q, and only deg 2 elements p (i.e., relations: (i), (ii), (iii) and (iii′) in (??)), since we are trying to prove that elements of deg ≥ 3 of GB are z-star.

• 2. Since q is z-star, the leading variable of the leading monomial LM
• f q is xαz with α the biggest 1st index in q.
• 3. In relation (i), out of all possible xijxij for different i, j, we are

restricted to xαzxαz but this one as a z-star element can’t result in non-star term when overlapping with z-star elements, and so relation (i) is ruled out.

• 4. In relation (ii), xijxkl − xklxij, we have j < l (consistent with our

lexicographic ordering). Therfore only xijxαz − xαzxij for j < z can

• verlap with z-star elements (and only on left of z-star term) to

make a non-star element.

• 5. In relation (iii), out of all xijxjk − xjkxik − xikxij for different i, j, k
• nly xiαxαz − xαzxiz − xizxiα and only when overlap on the left of

z-star element q, could result in non-star element.

Let T (d) be the set of deg(d) elements of a GB for FK(n). Let T (d+1)

II

be the set of deg(d + 1) elements of GB generated by

• verlap of elements of T (d) with degree 2 terms (relations type

(ii)) xijxkl − xklxij, where xij and xkl are generators of FK(n) such that {i, j} ∩ {k, l} = φ. Let for 3 ≤ z ≤ n, T (d) contain only z-star elements. Then T (d+1)

II

contains only z-star elements too.

Proof: Let g =

i Mi be an element of T (d), where the variables xξz and

xζz with different multiplicities could be factors of the monomials Mi for i = 1, 2, · · · , in different positions in monomials Mi. We calculate the overlap of relation xξζxαz − xαzxξζ with g =

i Mi,

where the left most variable of M1 is xαz, where α is the biggest 1st index in g. The corresponding syzygy is S = xξζ

• i Mi − (xξζxαz − xαzxξζ) M1

xαz

If we simplify the above and let S → P, we have P = xξζ

···

• i=2

Mi + xαzxξζ M1 xαz (2)

Now from P by applying Lemma we get the non-star component of P as follows. Pns = τξζ(

• i

Mi)xξζ = (τξζg)xξζ. (3) Now, if τξζg ∈ T (d) then (τξζg)xξζ is divisible by an element of GB and so it goes zero under reduction; therefore Pns = 0. Otherwise, If τξζg / ∈ T (d), then it is a linear combination of element of GB (by definition of GB), in which case again it goes zero under reduction and so Pns = 0. Therefore we have got no new non-star elements and we are left with only the z-star elements

By the Lemma, the z-star component of xξζM1 is Σk(M1)ξζ

k where

(M1)ξζ

k , by the 2nd part of Lemma, is made of M1 when the k-th

xξz(xζz) in M1, is followed by an xζz(xξz), respectively, where ξ ↔ ζ in all variables to the left of the k-th xξz(xζz), and the sign

• f (M1)ξζ

k is −(+) respectively.

Therefore the z-star component of P in the above will be calculated as follows. Ps = ···

i=2

• k(Mi)ξζ

k + xαz

• k( M1

xαz )ξζ k .

However xαz

• k( M1

xαz )ξζ k = k(M1)ξζ k (as {α, z} ∩ {ξ, ζ} = φ).

therefore Ps is simplified to Ps =

• i
• k

(Mi)ξζ

k .

(4) It is obvious that the above Ps is z-star

Let T (d) be the set of deg(d) elements of a GB for FK(n). Let T (d+1)

III

be the set of deg(d + 1) elements of GB generated by

• verlap of elements of T (d) with degree 2 terms (relations type

(iii)) xijxjk − xjkxik − xikxij, among the generators of FK(n). Let T (d) contain only z-star elements, for 3 ≤ z ≤ n. Then T (d+1)

III

contains only z-star elements too.

Proof: Let g =

i Mi be an element of T (d), where the variables

xξz and xζz with different multiplicities could be factors of the monomials Mi for i = 1, 2, · · · , in different positions in monomials Mi. We calculate the overlap of relation xξζxζz − xζzxξz − xξzxξζ with g =

i Mi, where the left most variable of M1 is xζz, where ζ is

the biggest 1st index in g. The corresponding syzygy is S = xξζ

• i Mi − (xξζxζz − xζzxξz − xξzxξζ) M1

xζz Therefore after

simplifying and letting S → P we have P = xξζ

···

• i=2

Mi + xζzxξz M1 xζz + xξzxξζ M1 xζz (5)

Now from P and by the Lemma we get the non-star component of P as following. Pns = (τξζ

• i

Mi)xξζ (6) From P by using 2nd part of the Lemma, the z-star elements are Ps =

···

• i=2
• k

(Mi)ξζ

k + xζzxξz

M1 xζz + xξz

• k

(M1 xζz )ξζ

k ,

(7) which is clearly a z-star polynomial.

Let T (d) be the set of deg(d) elements of a GB for FK(n). Let T (d+1)

III ′

be the set of deg(d + 1) elements of GB generated by

• verlap of elements of T (d) with degree 2 terms (relations type

(iii′)) xijxik − xjkxij + xikxjk, among the generators of FK(n). Let T (d) contain only z-star elements, for 3 ≤ z ≤ n. Then T (d+1)

III ′

contains only z-star elements too.

Proof: Let g =

i Mi be an element of T (d), where the variable

xζz with different multiplicities could be factors of the monomials Mi for i = 1, 2, · · · , in different positions in monomials Mi. We calculate the overlap of relation xζζ′xζz − xζ′zxζζ′ + xζzxζ′z with g =

i Mi, where the left most variable of M1 is xζz, where

ζ is the biggest 1st index in g. The corresponding syzygy is S = xζζ′

i Mi − (xζζ′xζz − xζ′zxζζ′ + xζzxζ′z) M1 xζz .

Under simplification and letting S → P we have P = xζζ′

···

• i=2

Mi + xζ′zxζζ′ M1 xζz − xζzxζ′z M1 xζz . (8)

Now from P and by the Lemma, we get the non-star component of P. Pns = (τζζ′

• i

Mi)xζζ′ (9) Also by 2nd part of the Lemma, the z-star elements are Ps =

···

• i=2
• k

(Mi)ζζ′

k

+ xζ′z(M1 xζz )ζζ′

k

− xζzxζ′z M1 xζz (10) which is clearly a z-star polynomial.

The above three propositions imply the following theorem.

Theorem

Let T (d) and T (d+1) be the sets of deg(d) and deg(d + 1) elements of a GB for FK(n) respectively. Let T (d) contain only z-star elements for 3 ≤ z ≤ n. Then T (d+1) contains only z-star elements for 3 ≤ z ≤ n too. Proof: According to the analysis in Remark ??, any possible non-star element of deg (d + 1) could be the result of overlap of elements of T (d) with relations ii, iii, and iii′, and already we checked all these three cases, in which no non-star elements generated.

Theorem

Let T (d) be the set of deg(d) elements of a GB for FK(n). Then, for d ≥ 3 , T (d) contains only z-star elements for 3 ≤ z ≤ n. Proof (by induction)

◮ Base of induction: Proposition 1 provides the base of induction

where it says the set of deg(3) elements of a GB for FK(n) contains only 3-star elements.

◮ Inductive Step: Theorem 3, provides the inductive step.