maximal degree among those bounded by Tα, Wγ (because, the bijection of Theorem 8.4.3 above is given by the map U 7→U ∪U#).
Theorem 8.4.5. M ulteβXαγ is the number of star sets U in β¯×β, which are of maximal degree among those which are chain-bounded by Teα and Wfγ.
Proof. Recall from [Kre08] that, if U is a multiset on β¯×β, then the monomial XU is square-free if and only if U is a subset of β¯×β, that is each of its elements has degree one. By Theorem 8.4.2, M ulteβXαγ is the number of square free monomials of maximal degree inP \in▷Gα,βγ . By Remark 8.4.4, this equals the number of ⋆⋆-multisets inβ¯×β, which are of maximal degree among those bounded by Tα, Wγ. Again, by Lemma 8.3.9, this equals the number of star sets in β¯×β, which are of maximal degree among those bounded byTα, Wγ. However, a subset ofβ¯×β is bounded by Tα, Wγ if and only if it is bounded by Teα and fWγ if and only if it is chain bounded by Teα and fWγ, where the last equivalence is due to (8.2.0.1).
8.5 Path families and multiplicities
In this subsection the write up is same as the write up of [Kre08].
For this subsection, let R and S be fixed positive and negative twisted chains contained inβ¯×β respectively. Let
MR= max{U ⊂( ¯β×β)− |R⊴Uand U is a star set}, MS = max {V ⊂( ¯β×β)+ | V ⊴S and V is a star set },
MSR = max {W ⊂( ¯β×β) | R⊴W− and W+⊴S and W is a star set },
where in each case by ‘max’ we mean the star sets U, V, or W respectively of maximal degree. For example, MSR consists of the collection of all star sets W of ( ¯β×β) which are of maximal degree among those which are chain bounded by R, S. When R = Teα and S=Wfγ,MSR consists precisely of the star sets U of Theorem 8.4.5. In order to give a better formulation of Theorem 8.4.5, we study the combinatorics ofMSR . Clearly,
MSR={U∪V˙ | U ∈ MR, V ∈ MS}, (8.5.0.1) where U∪V˙ is defined in (3.1.0.1) of Chapter 3. To study MSR, just like in [Kre08], we begin by consideringMR, and thus restricting attention to negative star sets of ( ¯β×β). Just like in [Kre08], a subset P ⊂ ( ¯β ×β)− is depth-one if it contains no two-element chains and ifP is depth-one, then it is anegative-path if the consecutive points are ‘as close as possible’ to each other, so that the points form a continuous path on ( ¯β×β)−
which moves only down or to the right. For any r= (e, f)∈( ¯β×β)−, define⌊r⌋and ⌈r⌉
are just like in [Kre08], that is
⌊r⌋= (e, f′), where f′ = min {y∈β | (e, y)∈( ¯β×β)−},
⌈r⌉= (e′, f), wheree′ = max {x∈β¯| (x, f)∈( ¯β×β)−}.
Now, we form the path Pr as follows:
1. it begins at⌊r⌋, and ends at⌈r⌉ and
2. if r = (e1, f1), r′ = (e′1, f1′) and r = (r′)#, then Pr =Pr′#. Also, if r = (e, f) and e=f⋆, then Pr is a negative star set in ( ¯β×β).
It is clear that if R is a twisted chain and if r′ ̸= r then Pr′ ∩Pr = ∅, where by ∅ we mean the empty intersection. Furthermore, R ⊴ S˙
r∈RPr. Define dR = P
r∈R|Pr|. Now, the following lemma is a straight forward consequence of the definitions.
Lemma 8.5.1. Let Q be a depth-one negative star set in β¯×β such that Pr ⊴Q. Then
|Q|≤ |Pr|, with equality if and only if Q is a negative-path from ⌊r⌋ to ⌈r⌉.
If U ⊂( ¯β×β)− is a star set,R ⊴U, and r ∈R, then define UR,r :={u∈U| r ⊴u, depthU(u) = depthR(r)}.
It follows from the definition that UR,r is depth-one. Indeed, if uand u′ are two elements of U which form a chain, then without loss of generality let u ≺ u′. Thus depthU(u) <
depthU(u′), and in particular, depthU(u)̸=depthU(u′). Thus u and u′ cannot both lie in UR,r.
Proof of Lemma 8.5.2 below is same as the proof of [Kre08, Lemma 10.2], so here we are omitting the proof.
Lemma 8.5.2. Let U ⊂( ¯β×β)− be a star set. Then 1. if R⊴U, then U = ˙S
r∈RUR,r.
2. If R ⊴ U, then |U|≤ dR, with equality if and only if U = ˙S
r∈RQr, where Qr is a negative-path from ⌊r⌋ to ⌈r⌉.
3. Let U = ˙S
r∈RQr ⊂ ( ¯β ×β), where Qr is a negative-path from ⌊r⌋ to ⌈r⌉. Then R⊴U.
Condition 2, of Lemma 8.5.2 implies that anyU ∈ MRis a disjoint unionU = ˙S
r∈RQr, where Qr is a negative-path from ⌊r⌋ to ⌈r⌉. Condition 3, of Lemma 8.5.2 implies that any disjoint union U = ˙S
r∈R, whereQr is a negative-path from ⌊r⌋ to⌈r⌉, is an element of MR. Consequently we have,
8.5. Path families and multiplicities 99
Corollary 8.5.3. MR consists of the set of all possible disjoint unions U = ˙S
r∈RQr, where Qr is a negative-path from ⌊r⌋ to ⌈r⌉.
Similar analysis can be done for positive star sets on β¯×β. Here the notion of a positive-path is as follows:
if P ∈ ( ¯β×β)+ is depth-one, then it is a positive-path if the consecutive points are ‘as close as possible’ to each other, so that the points form a continuous path on ( ¯β×β)+ which moves only up or to the left.
Likewise, the notions of ⌊r⌋ and ⌈r⌉ fors ∈( ¯β×β)+ can be defined as follows:
if r= (e, f)∈( ¯β×β)+, then we define
⌊r⌋= (e, f′), where f′ = max {y∈β |(e, y)∈( ¯β×β)+} and
⌈r⌉= (e′, f), where e′ = min {x∈β¯| (x, f)∈( ¯β×β)+}.
Corollary 8.5.4. MS consists of the set of all possible disjoint unions V = ˙S
s∈SQs, where Qs is a positive-path from ⌊s⌋ to ⌈s⌉.
Corollary 8.5.3, Corollary 8.5.4, and (8.5.0.1) together imply the following corollary.
Corollary 8.5.5. MSR consists of the set of all possible disjoint unions W = ˙S
r∈R∪SQr, whereQr is either a negative-path or a positive-path from ⌊r⌋to ⌈r⌉depending on whether r is negative or positive.
The star sets U of Theorem 8.4.5 are precisely the elements of MSR, when R = ˜Tα and S = ˜Wγ. Therefore, combining Theorem 8.4.5 and Corollary 8.5.5, we obtain the following theorem.
Theorem 8.5.6. M ulteβXαγ is the number of disjoint unions S˙
r∈fTα∪fWγPr, where Pr is either negative-path or a positive-path from⌊r⌋to ⌈r⌉, depending on whetherr is negative or positive.
Example 8.5.7. Let d= 5, that is, 2d= 10. Let α= (1,2,4,6,8), β = (2,4,5,8,10), γ = (3,5,7,9,10). Clearly, α, β, γ ∈ I(d), and α ≤ β ≤ γ. Again, as α, β, γ ∈ I(d) so by Lemma 8.3.7, Teα,fWγ both are star sets. We want to compute M ulteβXαγ. The following two diagrams show the negative and positive twisted chains Teα = {r1, r2} and Wfγ = {s1, s2, s3} in β¯×β; and the set of ⌊r⌋’s and ⌈r⌉’s for all r ∈ Teα ∪Wfγ. Note that, s1 =⌊s1⌋=⌈s1⌉ and s3 =⌊s3⌋=⌈s3⌉.
p p p p p
p p p p p
p p p p p
p p p p p
p p p p p
9 7 6 3 1
2 4 5 8 10
r1
r2 s1
s2
s3 p
p p p p
p p p p p
p p p p p
p p p p p
p p p p p
9 7 6 3 1
2 4 5 8 10
⌊s1⌋
⌈s1⌉
⌊s3⌋
⌈s3⌉
⌈s2⌉
⌊s2⌋
⌊r1⌋
⌈r1⌉
⌈r2⌉
⌊r2⌋
−→boundary of N(β)
There are four non intersecting path families from ⌊r⌋ to ⌈r⌉, r ∈ Teα ∪Wfγ, as shown below. Thus M ulteβXαγ = 4.
p p p p p
p p p p p
p p p p p
p p p p p
p p p p p
9 7 6 3 1
2 4 5 8 10
•
• p
p p p p
p p p p p
p p p p p
p p p p p
p p p p p
9 7 6 3 1
2 4 5 8 10
•
•
p p p p p
p p p p p
p p p p p
p p p p p
p p p p p
9 7 6 3 1
2 4 5 8 10
•
• p
p p p p
p p p p p
p p p p p
p p p p p
p p p p p
9 7 6 3 1
2 4 5 8 10
•
•
CHAPTER 9
MULTIPLICITY AT A TORUS-FIXED POINT IN A RICHARDSON VARIETY IN THE ORTHOGONAL GRASSMANNIAN
A Richardson variety is the intersection of a Schubert variety and an opposite Schubert variety. In [Upa08], Raghavan and Upadhyay have already found the multiplicity at any torus-fixed point in a Schubert variety in the orthogonal Grassmannian. In this chapter, we define and elaborate everything only for the opposite Schubert variety in the orthogonal Grassmannian and then merge the two results.
For the rest of this chapter, let d be a positive integer and w′ ≤v ≤wbe elements of OI(d). Also, recall the notation O(R(v)e \N(v))from §4.1.
9.1 Vertical and horizontal projections of an element α in O(R(v) e \ N(v))
Letα = (r, c)be an element inO(R(v)e \N(v)), then the elementpv(α) := (c⋆, c)is called the vertical projection of α and ph(α) := (r, r⋆) is called the horizontal projection ofα. The lines joiningα to its projections are called the legsof α. For a monomial C of R(v), we define
C(down)=C∩(OR(v)e ∪d(v)).
Example 9.1.1. Let d= 7 and v = (4,6,7,10,12,13,14). Let the monomial S={(9,10)2,(8,14),(11,6),(5,7),(2,12)2}.
Then
S(down)={(9,10)2,(8,14),(11,6)}.