The grid representing R(v)
A v -chain
A monomial S
The monomial S and its block decomposition
Left concatenation of a block
The monomial F and its block decomposition
The block decomposition of the monomial F
The block decomposition of the monomial F (1)
The block decomposition of the monomial U
The block decomposition of the monomial U (1)
Chain and antichain
A τ -line
Illustration of the grid representing N(v)
The monomials S w ′ and T
The monomial S
The monomials S pr 1,2 and S pr 3,4
The grid representing the monomials S 1,2 and S 3,4
The monomials S ′ 1,2 and S ′ 3,4
An element of Paths w w ′
Another element of Paths w w ′
Introduction
Organization of the thesis
In Chapters 3 and 4, we described the limited RSK correspondence (provided by Kreiman in [Kre08]) and the Kodiyalam-Raghavan maps (provided in [KR03]). In chapters 8 and 9, we gave the formula for the multiplicity of the Richardson variety at any fixed torus point in the Symplectic or orthogonal Grassmannian.
A comment about the figures
To begin with, Chapter 2 described our basic objects of interest, namely, symptomatic and orthogonal Grassmannians, and Richardson varieties on them. In Chapter 7, we used the main result of Chapter 5 to calculate Gröbner's explicit basis.
The orthogonal Grassmannian
We denote by Md(V) the orbit of the space of e1,· · ·, and call it even orthogonal.
Richardson varieties in the orthogonal (symplectic) Grassmannian
The Zariski closure of the B+ orbit (respectively B−-orbit) through eα, with the canonical structure of the reduced scheme, is called a Schubert variety (respectively opposite to the Schubert variety), and is denoted by Xα (respectively every B+ orbit (respectively B−-orbit) by was irreducible and open in its closure, it follows that the closures of the B+ orbit (ref.
The RSK correspondence
Young tableau and ordinary insertion
Ordinary RSK correspondence
The bounded RSK correspondence
- Notched tableaux
- The bounded insertion
- Semistandard notched bitableaux
- The bounded RSK correspondence
Again, points that are to the left of the boundary of N(v), and that also lie above the diagonal, are the points of ON(v). Points to the right of the boundary of N(v) and below the diagonal belong to the set O(R(v)e \N(v)).
Description of the map π
Some necessary results
Then there exists an element (R, C) of the noble monomial Sw associated with w such that C ≤c and r≤R. For a positive integer j, let Sjw denote the subset of Sw of the elements that are j-deep, and wj the corresponding element of I(d, N).
Description of the map ϕ
Statement of the main theorem
Let U be a finite monomial in N(v). i) all row numbers of the distinct subset Swr (corresponding to wr) consist of the ((m+ 1)−r)th row entries of the left cut table BRSK(U). ii) All column numbers Swr consist of the entries of the ((m+ 1)−r)th row of the right notched table BRSK(U). The same applies to the column numbers Sw1, which comprise the entries of the th row of the right notched table BRSK(U). But it follows from the statement of Theorem 5.1.1 that all the row numbers of a distinct subset Swr (the corresponding column) consist of P(m+1)−r entries, and all the column numbers of Swr consist of Q(m+1)− entries d.
Now, if we look at the proof of Remark 4.1.13 (the second paragraph of the proof, to be more precise), the procedure given there shows that. numbers of rows of Swr)∪· (v\numbers of columns ofSwr) = wr.
The strategy and the proof of Theorem 5.1.1
The dark circles indicate the elements of the monomial S, and the numbers written near these dark circles indicate the elements of the monomial S. The dark circles in Figure 5.2 indicate the elements of the monomial S∪ { (8,4)}, and the numbers written near these dark circles indicate the abundance of these elements in the monomial S∪ {(8,4)}. Then there exists a block B of F such that while forming the monomial U from the monomial F, the element (bn, an) is added to block B of F, but not to the leftmost end.
This is simply because bn is the smallest possible row number among all elements of U.
Lemmas needed to prove Theorem 5.1.8
- A general lemma
- Lemma for Case I
- Lemmas for Case II
- An illustration of Lemmas 5.2.5 and 5.2.7 for the case k ′ = 0
- For any monomial U in R(v) \ N(v) , π(U ˜ ) = BRSK(U )
If ek ≤r1, then indeed (r1, c1) is not the first element of the highest block of F of depth k, a contradiction. Note that 1 and 2 are the column numbers of the first elements of some blocks of depth < k = 3. Also, b < c1 implies that the smallest column number of blockB is less than or equal to b (which is less than c1), a contradiction.
Then there is at least one element of the form (bn, p) in D, where p̸=c1, and there are no elements in D that lie between (bn, p) and (bn, c1).
An application of the main theorem
- Some necessary definitions and notation
- The result of Ghorpade and Raghavan
- The application
- Proof of the application
- Proof of the equivalence of the bijections of Kreiman’s Thesis (
Given v and w in I(d), we denote by SMwv the set of all standard sequences compatible with v dominated by w. Let SMv,v denote the set of all standard sequences compatible with v that are anti-dominated by v: a standard sequence w1. Elements of the form (r, r⋆) of N(v) are referred to as belonging to the "diagonal", and the set of all diagonal elements of N(v) is denoted med(v). 2) The set of every diagonal element in S is even.
The rest of the proof now follows from the definition of Swτ(m) (in the sense of §5.3.1).
Gröbner basis
Ideals of tangent cones to Richardson varieties
The affine spot Aβ :=Md(V)∩A of the symplectic Grassmannian Md(V) is an affine space whose coordinate ring can be understood as the polynomial ring in variables of the form X(r,c) with (r, c )∈OR(β) (mentioned in §4.1). For θ ∈I(d,2d), consider the submatrix of the above matrix, given by the rows numbered θ\β and columns numbered β \θ. From [LMS79] we can derive a series of generators for the ideal Iα,βγ of functions onAβ that vanish onYαγ(β). We are interested in the tangent cone to Xαγ ateβ or, which is the same, the tangent cone toYαγ(β) at the origin .
Because of this, Yaγ(β) is itself a cone and equal to its tangent cone at the origin.
Extended β -chains
7.3.0.3) We are interested in the tangent cone to Xαγ ateβ or, what is the same, the tangent cone to Yαγ(β) at the origin. The proof of the fact that bot(C−) belongs to I(d) is similar (we omit the proof here because it involves the proof that the maps BRSK and π˜ are equal on negative multisets, and this proof is similar to that in Chapter 5).
Gr o ¨ bner basis for ideals of tangent cones
Strategy of the proof
Similarly, we have to prove that the number of monomials of P \in▷Gα,βγ is to some extent not greater than the number of monomials of P \in▷Iα,βγ. Therefore, it is sufficient to prove that the number of monomials of P \in▷Gα,βγ is less than or equal to the number of standard monomials of Yαγ(β) in any degree. In §7.7 below, we will first show that there exists a degree-doubling injection from the set of all monomials of P\in▷Gα,βγ to the previous set.
We will then show that there is a degree-halving injection from the posterior set (namely the set of all “non-vanishing semi-standard notched bitableaux on ( ¯β×β)∗ (bounded byTα, Wγ)”) to the set of all standard monomials on Yαγ(β).
The two sets
The first set
The second set
The proof
There is a degree-doubling injection from the set of all monomials from P \in▷Gα,βγ to the set of all nonextending special multisets on β¯×β (bounded by Tα, Wγ). There is a graded halved injection from the set of all non-vanishing semi-standard notched bitables on ( ¯β×β)∗ (bounded by Tα, Wγ) into the set of all standard monomials on Yαγ(β). The BRSK mapping from Chapter 3 is a degree-preserving bijection from the set of all nonzero special multisets on β¯×β (bounded by Tα, Wγ) to the set of all nonzero semistandard notched bitables on ( ¯β×β)* (bounded by Tα, Wγ).
Therefore, we can now conclude that mapBRSK in Chapter 3 is a degree-preserving bijection from the set of all positive special multisets onβ¯×β (bounded by ∅,Wγ) to the set of all positive semistandard notched bitableaux on( ¯β ×β)∗ ( bounded by ∅, Wγ).
Multiplicity
We will now define what is meant by the multiplicity of a maximal ideal on a local ring. The leading coefficient PM,R is of the form e(M,R). dimR denotes the dimension of the local ring R) for some positive integer e(M, R). Let α, β and γ be the elements of I(d), which were determined at the end of Chapter 2.
In our case, R is the local ring OXαγ,eβ of germs at eβ of function on the Richardson manifold Xαγ, and M is its unique maximal ideal Meβ.
Some necessary definitions and notation
We define the associated graded ring of A with respect to I, written grIA to be the graded ring.
Some necessary definitions and lemmas
There exists a bijection between elementswofI(d, N)satisfyingw≥v on the one hand and subsets S of N(v) satisfying both conditions for distinguished subset (Definition 4.1.11). It is enough to show that fWγ is a star set in β¯×β (for Teα the proof is the same). There exists a bijection from the set of all ⋆⋆ multisets in β¯×β to the set of all star sets in β¯×β.
Consider the map ψ from the set of all ⋆⋆-manifolds in β¯×β to the set of all star groups in β¯×β given by ψ(U) = fundamental set of U .
The main theorem
Path families and multiplicities
If u and u′ are two elements of U that form a chain, then let u ≺ u′ without loss of generality. In [Upa08], Raghavan and Upadhyay have already found the manifold on any torus-fixed point of a Schubert variety in the orthogonal Grassmannian. In this chapter, we only define and elaborate everything for the opposite Schubert variant 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).
Definition of anti- O -domination
The main theorem of this chapter
Reduction of the proof to combinatorics
Standard monomial theory
Notation: given v, w and w′ ∈OI(d), we denote by SMww′(v) the set of all w-dominated, w′-anti-dominated and v-compatible standard monomials. The standard monomials qθ1· · ·qθr of degree r form the basis for the space of degree forms in the homogeneous coordinate ring Md(V) in the embedding defined by the extensive generator L of the Picard group. More generally, for w, w′ ∈ OI(d), the standard monomials of degree r, dominant and dominated by w′, form a basis for the space of forms of degree r in the homogeneous coordinate ring of the Richardson variety Xww ′ of Md(V).
To prove Theorem 9.5.1 it is therefore sufficient to show that the set SMww′(v)(m) of w-dominated w′-anti-dominated v-compatible standard monomials of degree m is in bijection with Sww′( v)( m), as stated in Theorem 9.6.4 below.
The proof
The theorem above tells us that the graded piece degree of K[Yww′(v)] is generated as a K-vector space by elements of SMww′(v) of degree m, where the degree of a standard monomial fθ1· · · fθt is defined as the sum of the v-degrees of θ1,· · ·, θt. The setSMww'(v)(m) of standard monomials in OI(d) of degree m which are v-compatible, dominated by w, and anti-dominated by w' is in conjunction with the set Sww′(v)(m) ) of monomials in ON(v)∪O(R(v)e \N(v)) of degree m that are O-dominated by w as well as anti-O-dominated by w′. For that we will first show that there exists a bijection between the sets SMwv′(v)(m) (where SMwv′(v)(m) denotes the set of all standard monomials in OI(d) of degree m that v -compatible, dominated by v, and anti-dominated by w′) and Uw′(m), where Uw′ denotes the set of all monomials in O(R(ve )\ N(v)) that is anti -O -dominated by w′ and Uw′(m) denote such monomials of degree m.
Now we know from [Upa08] that there is a bijection between the sets SMvw(v)(m) and Tw(m), where Tw denotes the set of all monomials in ON(v) that are O-dominated by w and Tw (m) denotes such monomials of degree m.
The type of an element α in an anti- v -chain C
O -depth of an element in a monomial in O(R(v) e \ N(v))
Description of the map Oπ
Description of the map Oϕ for monomials in O(R(v)e \N(v)) 109wherew′ and S′ satisfy the conditions of proposal 9.7.1.
Description of the map Oϕ for monomials in O(R(v) e \ N(v))
Some important lemmas
Then there exists an element (R, C) in the distinguished monomial Sw′ associated with w′ such that R ≤r and C ≥c. For a positive integer j, let Sjw′ denote the subset of Sw′ of those elements that are j-deep, and w′j the corresponding element of I(d,2d).
Description of Oϕ
Two separate elements belonging to the same Tjw′ are not comparable: Let β, β′ be both elements of Tjw′ such that β > β′. Since no two distinct elements of Tjw′ are comparable, the column entries are also in non-increasing order. In fact, the number of diagonal elements of Sw′,j,j+1 is either 0 or 2 (in the latter case the elements must be distinct, since Sw′ is distinct and therefore multiplicity free).
In Figure 9.2, the open circles indicate the points of Sw′, the dark circles indicate the points of T, and di indicates the depth of the corresponding elements of Sw′ and O-depths of the elements of T.
Multiplicity counts using certain lattice paths
Description and illustration
The number of such p-tuples, where p:=|Sw′(up)∪Sw′(down)|set Xww′.
Justification for the interpretation
In [AIJK20], Ikeda and his co-authors calculated the multiplicity of any Schubert variety in the symplectic flag variety. Schubert varieties in the Grassmannian and the symplectic Grassmannian via a bounded RSK correspondence (Published) (Indian Journal of Pure and Applied Mathematics, DOI: 10.1007/s. Multiplicity at any torus-fixed point in a Richardson variety in the symplectic Grassivmannian, Submitted ).
Local properties of Richardson varieties in Grassman via a restricted Robinson-Schensted-Knuth correspondence.
