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 x ij = − x ji for 1 ≤ i < j ≤ n with the following relations: i : x 2 for i < j ; ij ii : x ij x kl − x kl x ij = 0 whenever { i , j } ∩ { k , l } = φ, i < j , and k < l ; iii : x ij x jk − x jk x ik − x ik x ij = 0 , iii’ : x ij x ik − x jk x ij + x ik x jk = 0 for i < j < k . (1)

In other words, Let V be the vector space spanned by the generators x ij . Then FK ( n ) is a quotient of the tensor algebra T ( V ) = ⊕ n ≥ 0 V ⊗ n , the free associative algebra on the generators of 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 [ x 1 , · · · , x n ] is a finite set of generators { g 1 , · · · , g m } for I whose leading terms generate the ideal of all leading terms in I , i.e., I = ( g 1 , · · · , g m ) and LT ( I ) = ( LT ( g 1 ) , · · · , LT ( g m )).

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 mm 1 ≥ mm 2 for monomials m , m 1 , m 2 . Equivalently, a monomial ordering may be specified by defining a well ordering on the n-tuple α = ( a 1 , · · · , a n ) ∈ Z n of multi-degrees of monomials Ax a 1 1 · · · x a n 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 M 1 is smaller than that of monomial M 2 , then M 1 comes first, M 1 < M 2 , 2. if the 2nd indexes of the 1st variables of M 1 and M 2 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 M 1 and M 2 happens to be of the same order, then look at the next variable. Example x 23 x 13 x 23 < x 14 x 23 x 13 , x 23 x 13 x 23 < x 13 x 23 x 13 , x 23 x 24 < x 23 x 14 .

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 [ x 1 , · · · , x n ] be nonzero polynomials. 1. If multideg ( f ) = α and multideg ( g ) = β , then let γ = ( γ 1 , · · · , γ n ), where, γ i = max { α i , β i } for each i . We call x γ the ”least common multiple” of LM ( f ) and LM ( g ), written x γ = LCM ( LM ( f ) , LM ( g )). 2. The S − polynomial of f and g is the combination x γ x γ S ( f , g ) = LM ( f ) f − LM ( g ) g .

Definition ”Reduction of S-polynomial S ( f , g ) with respect to the list of G is defined as generators G = [ g 1 , g 2 , · · · , g t ]”, denoted S ( f , g ) 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 = < f 1 , · · · , f s > � = { 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 = ( f 1 , · · · , f s ) Output: a Grobner basis G = ( g 1 , · · · , g t ) for I , with F ⊂ G G := F REPEAT G ′ : G FOR each pair { p , q } , p � = q in G ′ DO G ′ S := S ( p , q ) IF S � = 0 THEN G := G ∪ S UNTIL G = G ′ G ′ In the above, S ( p , q ) is the remainder of division of S − polynomial S ( p , q ) by the ordered elements of G ′ . We call G ′ ”reduction of S(p,q) with respect to the list G ′ ”. S ( p , q )

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 x α z , where 1 ≤ α < z . A homogenous polynomial where all monomials are z-star with the same z , is called a z-star polynomial. Example x 23 x 13 x 23 + x 13 x 23 x 13 is a 3-star polynomial, while x 23 x 12 x 23 + x 13 x 23 x 13 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 x ij a generator of FK ( n ) such that i , j � = z. Then, under the reduction algorithm, x ij M, is splitted into one or more z-star monomials (with the same z) and a non-star monomial, i.e., x ij M → a non-star monomial + a z-star polynomial. Moreover 1. The non-star monomial is equal to [ τ ij M ] x ij , where τ ij is a transposition in the group of permutations. k ( M ) ij 2. The z-star polynomial is � k for k = 1 , 2 , 3 , · · · , where ( M ) ij k is made of M as following. Let M = x a 1 z x a 2 z · · · x a k z · · · x a d z . if a k � = i , j then ( M ) ij For k = 1 , · · · , d , k = 0 ; otherwise if then ( M ) ij M � � a k = i , k = − τ ij ( x a 1 z · · · x a k − 1 z ) x iz x jz ( x a 1 z ··· x ak z ) ; then ( M ) ij � � M if a k = j , k = τ ij ( x a 1 z · · · x a k − 1 z ) x jz x iz ( x a 1 z ··· x ak z )

The set of degree 3 elements of a GB for FK ( n ) is T (3) = g (3) = x bz x az x bz + x az x bz x az : 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, x ab x cz − x cz x ab with x cz x cz yields P = x cz x ab x cz that under the algorithm it reduces to zero. Overlap between relations ( ii ) and ( iii ) results in zero as follows x αβ x ab − x ab x αβ for 1 ≤ α < β < b < z ≤ n and a < b with x ab x bz − x bz x az − x az x ab gives P = − x αβ x bz x az − x αβ x az x ab + x ab x αβ x bz which goes zero under reduction algorithm. Also overlap of x ab x bz − x bz x az − x az x ab with x bz x cz ′ − x cz ′ x bz for z < z ′ results in P = − x ab x cz ′ x bz + x bz x az x cz ′ + x az x ab x cz ′ 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 G ′ S ( p , q ) 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 ( x ij x α z − x α z x ij , q ), S ( x i α x α z − x α z x iz − x iz x i α , q ), and S ( x i α x iz − x α z x i α + x iz x α z , 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 of q is x α z with α the biggest 1st index in q . 3. In relation ( i ), out of all possible x ij x ij for different i , j , we are restricted to x α z x α z 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 ), x ij x kl − x kl x ij , we have j < l (consistent with our lexicographic ordering). Therfore only x ij x α z − x α z x ij for j < z can overlap 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 x ij x jk − x jk x ik − x ik x ij for different i , j , k only x i α x α z − x α z x iz − x iz x i α and only when overlap on the left of z-star element q , could result in non-star element.