We now specify theterm order▷on monomials in the coordinate functions{X(r,c)|(r, c)∈ OR(β)} with respect to which the initial ideal of the ideal Iα,βγ of the tangent cone is to be taken.
Definition 7.5.1. Let > be the total order on OR(β) satisfying the following condition:
• X(r,c) > X(r′,c′) if either (a) r > r′ or (b) r=r′ and c < c′.
Let ▷ be the term order on monomials in OR(β) given by the homogeneous lexicographic order with respect to >.
Example 7.5.2 below gives an illustration of the term order ▷.
Example 7.5.2. Letd= 7andβ = (1,3,4,7,9,10,13). Now, all of(14,1),(12,3),(11,3), (11,1) are elements in OR(β), and according to the above term order we have, X(14,1) >
X(12,3) > X(11,1) > X(11,3). LetXS1 =X(14,1)3 X(11,1)X(11,3)2 , XS2 =X(14,1)X(12,3)X(11,3), and XS3 = X(14,1)X(11,1)2 . Clearly, S1, S2, and S3 all are monomials in OR(β). Now, the degree of the polynomial XS1 is greater than that of XS2 and XS3. So, XS1 ▷ XS2 and XS1 ▷XS3. Though the degree of XS2 is equal to the degree of XS3, but X(12,3) > X(11,1) and in XS2, the degree of X(12,3) is one and in XS3, the degree of X(12,3) is zero. So, ac- cording to the definition of homogeneous lexicographic order, we have XS2 ▷XS3. Hence, XS1 ▷XS2 ▷XS3.
Now, recall that the ideal of the tangent cone to Xαγ at eβ is the ideal Iα,βγ given by (7.3.0.3). Let ▷ be as in §7.5. For any element f ∈Iα,βγ , let in▷f denote the initial term of f with respect to the term order ▷. We definein▷Iα,βγ to be the ideal⟨in▷f |f ∈Iα,βγ ⟩ inside the polynomial ring P :=K[X(r,c) | (r, c)∈OR(β)].
Definition 7.5.3. An admissible pair w= (t, u) (where t ≥u) is called a good admis- sible pair if it satisfies both of the following 2 properties:
1. α≰u or t≰γ.
2. Eitherin▷fw,β forms a positive upper extendedβ-chainC+ such thatC(1)+ −C(2)+ ≰γ or in▷fw,β forms a negative upper extended β-chain C− such that C(1)− −C(2)− ≱α.
Notation: LetGα,βγ denote the set {fw,β | wis good}. Example 7.5.4 below illustrates a good admissible pair.
Example 7.5.4. Let d = 4, α = (1,2,3,5), β = (1,2,5,6), and γ = (2,3,5,8). Let w= (t, u) be an admissible pair, where t= (3,4,7,8) and u = (1,2,5,6). Clearly, t ≰γ.
Now, in § 7.3, we have already defined that θ = (t ∩[d])∪(u∩[d]c), and fw,β = fθ,β.
7.5. Gr¨obner basis for ideals of tangent cones 79
Hence, in this exampleθ = (3,4,5,6) and from the matrix which is given in (7.3.0.1), we have
fw,β =
x31 x32
x41 x42
(7.5.0.1) Observe thatin▷fw,β =−x41x32. Clearly,in▷fw,β forms a positive upper extendedβ-chain C+ such that C(1)+ −C(2)+ ={3,4} ∪(β∖{1,2}) = (3,4,5,6)≰γ. Hence, w = (t, u) is a good admissible pair.
Definition 7.5.5.IfSis any nonempty subset of the polynomial ringP :=K[X(r,c)|(r, c)∈ OR(β)] such that S ̸={0}. We define in▷S to be the ideal ⟨in▷(s) | s ∈S⟩.
The main result of this chapter is the following:
Theorem 7.5.6. The set Gα,βγ is a Gröbner basis for the ideal Iα,βγ .
7.5.1 Strategy of the proof
To explain the strategy of the proof of Theorem 7.5.6, we need the following definition.
Definition 7.5.7. We call f = fw1,β· · ·fwr,β ∈ P =K[X(r,c) | (r, c) ∈ OR(β)] a stan- dard monomial if
w1 ≤ · · · ≤wr, (7.5.1.1)
and for each i∈ {1, . . . , r}, we have
Either β ≥top(wi) or top(wi)≥β, (7.5.1.2) and either bot(wi)≥β or β ≥bot(wi) (7.5.1.3)
and wi ̸= (β, β). (7.5.1.4)
If in addition, forα, γ ∈I(d), we have
α ≤bot(w1) and top(wr)≤γ, (7.5.1.5) then we say thatf is standard on Yαγ(β).
Example 7.5.8 below gives an illustration of a standard monomial on Yαγ(β).
Example 7.5.8. Let d = 4, α = (1,2,3,5), β = (1,2,5,6), and γ = (3,4,7,8). For this β, the 8×4 matrix is given by (7.3.0.1). Let w1 = ((1,2,4,6),(1,2,3,5)) and w2 = ((2,4,6,8),(2,3,5,8)). Clearly, w1 andw2 both are admissible pairs. Letθ1 and θ2 be the images of w1 and w2 respectively, under the correspondence given by w = (x, y) 7→ θ = (x∩[d])∪(y∩[d]c)as mentioned in [GR06, Proposition 3.4]. So, we have, θ1 = (1,2,4,5) and θ2 = (2,4,5,8). Now, using the 8×4 matrix of (7.3.0.1) we have,
fw1,β =fθ1,β =x35∈P
and fw2,β =fθ2,β =−x41x31−x35x81 ∈P.
Hence, f = fw1,βfw2,β ∈ P. Clearly, w1 ≤ w2. Again, β ≥ top(w1) and β ≥ bot(w1).
Also, β ≤ top(w2) and β ≤ bot(w2), and wi ̸= (β, β) for all i ∈ {1,2}. Again, α, γ ∈I(d) are such that α≤bot(w1) and γ ≥top(w2). Hence, f is standard on Yαγ(β).
Definition 7.5.9. Let f = fw1,β· · ·fwr,β be a standard monomial on Yαγ(β). We define the degree of f to be the sum of the β-degrees of w1, . . . ,wr, where given any admissible pair w= (x, y), the β-degree of w is defined to be 12(|x\β|+|y\β|).
We now briefly sketch the proof of Theorem 7.5.6 (the details can be found in §7.7).
Clearly,Gα,βγ is contained in the ideal Iα,βγ . So, in▷Gα,βγ ⊆in▷Iα,βγ . Hence, to prove Theo- rem 7.5.6, we only need to show that in any degree, the number of monomials of in▷Gα,βγ is at least as great as the number of monomials of in▷Iα,βγ (the other inequality being trivial). Equivalently, we need to prove that in any degree, the number of monomials of P \in▷Gα,βγ is no greater than the number of monomials of P \in▷Iα,βγ . Both the mono- mials of P \in▷Iα,βγ and the standard monomials on Yαγ(β) (the definition of a standard monomial onYαγ(β)is given in Definition 7.5.7) form a basis forP/Iα,βγ , and thus agree in cardinality in any degree. Therefore it suffices to prove that, in any degree, the number of monomials ofP \in▷Gα,βγ is less than or equal to the number of standard monomials on Yαγ(β). In this chapter, we consider two sets, namely, the set of all “nonvanishing special multisets on β¯×β (bounded by Tα,Wγ)”, and the set of all “nonvanishing semistandard notched bitableaux on ( ¯β×β)∗ (bounded by Tα, Wγ)”. The meaning attached to these two sets is given in §7.6 below. In §7.7 below, we will first show that there exists a degree doubling injection from the set of all monomials ofP\in▷Gα,βγ to the former set. Then we will show that, there exists a degree-halving injection from the later set (namely, the set of all “nonvanishing semistandard notched bitableaux on ( ¯β×β)∗ (bounded byTα, Wγ)”) to the set of all standard monomials on Yαγ(β). And then we will prove that the map BRSK of Chapter 5 is a degree preserving bijection from the former set to the later. This will complete the proof.
Example 7.5.10 below gives an illustration of a Gro¨bner basis.
Example 7.5.10. Let d = 4, α = (1,2,3,5), β = (1,2,5,6), and γ = (2,3,5,8). Then from Example 7.5.4, we know that, w = (t, u) is a good admissible pair, where t = (3,4,7,8)and u= (1,2,5,6). For the above α, β, γ if we consider all the admissible pairs which satisfying both the conditions of good admissible pair, then we will get the set Gα,βγ which is given by{fw,β|w∈G}, whereGis the following set (of all good admissible pairs):
G={((1,2,3,4),(1,2,3,4)),((1,4,6,7),(1,2,5,6)),((2,4,6,8),(1,2,5,6)),((3,4,7,8),(1 ,2,5,6)),((1,5,6,7),(1,5,6,7)),((2,5,6,8),(1,5,6,7)),((3,5,7,8),(1,5,6,7)),((4,6,7,8) ,(1,5,6,7)),((2,5,6,8),(2,5,6,8)),((3,5,7,8),(2,5,6,8)),((4,6,7,8),(2,5,6,8)),((5,6,7