Applications of Jeu de Taquin to Representation Theory and Schubert Calculus

Table of contents :
1 Title i
2 Abstract ii
3 Symmetric Functions 1
3.1 The Ring of Symmetric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
3.2 Schur Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4 Applications to Representation Theory and Schubert Calculus 7
4.1 The Representation Theory of S n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 Representation Theory of GL n (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Schubert Calculus on the Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Jeu de Taquin and the Littlewood-Richardson Rule 13
5.1 Jeu de Taquin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 The Littlewood-Richardson Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.3 Growth Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Beyond Type A Grassmannians 18
6.1 (Co)Minuscule Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.2 A Jeu de Taquin Procedure for (Co)Minuscule Flag Varieties . . . . . . . . . . . . . . . . . . 20
6.3 Example: Root Systems of Type A n−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.4 Example: Root Systems of Type B n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.5 Example: Root Systems of Type C n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.6 Example: Root Systems of Type D n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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Applications of Jeu de Taquin to Representation Theory and Schubert Calculus by

Daniel S. Hono II

A Thesis Submitted to the University at Albany, State University of New York in Partial Fulfillment of the Requirements of the Degree of Master of Arts

College of Arts and Sciences Department of Mathematics and Statistics 2019



   

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Abstract We describe some applications of the jeu de taquin algorithm on standard Young tableaux of skew shape λ/µ for λ and µ partitions. We first briefly survey the relevant background on symmetric functions with a focus on the Schur functions. We then introduce the Littlewood-Richardson coefficients in terms of Schur functions and survey some of the applications of the corresponding Littlewood-Richardson rule to representation theory and Schubert calculus on the Grassmannian. A reformulation of the LittlewoodRichardson rule in terms of the jeu de taquin algorithm and growth diagrams is then surveyed. We illustrate this formulation with some examples. Finally, we discuss some work by Hugh Thomas and Alexander Yong in extending the Littlewood-Richardson rule to the more general setting of (co)minuscule flag varieties. In this setting we describe another reformulation of growth diagrams in terms of chains in Bruhat order and describe some examples of the generalized jeu de taquin for root systems of types An−1 , Bn , Cn , and Dn .

ii

Contents 1 Title

i

2 Abstract

ii

3 Symmetric Functions

1

3.1

The Ring of Symmetric Functions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

3.2

Schur Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

4 Applications to Representation Theory and Schubert Calculus

7

4.1

The Representation Theory of Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

4.2

Representation Theory of GLn (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

4.3

Schubert Calculus on the Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

5 Jeu de Taquin and the Littlewood-Richardson Rule

13

5.1

Jeu de Taquin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

5.2

The Littlewood-Richardson Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

5.3

Growth Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

6 Beyond Type A Grassmannians

18

6.1

(Co)Minuscule Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

6.2

A Jeu de Taquin Procedure for (Co)Minuscule Flag Varieties . . . . . . . . . . . . . . . . . .

20

6.3

Example: Root Systems of Type An−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

6.4

Example: Root Systems of Type Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

6.5

Example: Root Systems of Type Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

6.6

Example: Root Systems of Type Dn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

iii

Chapter 3

Symmetric Functions We begin our exposition with the definition of the ring of symmetric functions Λ. We will then explore some of the properties of the functions that form a basis of Λ. These functions are the monomial, elementary, and complete homogeneous symmetric functions. Only a brief overview of their properties are given. In the next section, we then define a remarkable basis for Λ known as the Schur functions. The Schur functions play important roles in the representation theory of the symmetric group, the representation of GLn , and the computation of the cup product in the cohomology ring of the Grassmanian. The Schur functions arise in many different areas and therefore can be defined in a variety of ways. We shall define them in a combinatorial manner using semi-standard Young Tableaux (SSYT). Recall, a symmetric polynomial (with integer coefficients) is a polynomial p ∈ Z[x1 , . . . xn ] such that, for every σ ∈ Sn we have that p(xσ(1) , . . . , xσ(n) ) = p(x1 , . . . , xn ). That is, permuting the indeterminates x1 , . . . xn does not change the polynomial. Symmetric functions are then a generalization of symmetric polynomials in which the number of indeterminates is infinite. We begin with symmetric functions and then we will specialize to the case of symmetric polynomials wherever necessary. In the following, let P ar be the set of all integer partitions and let P ar(n) denote the set of all partitions of the positive integer n.

3.1

The Ring of Symmetric Functions

Let n ∈ N, then we call a sequence α = (α1 , α2 , . . .) a weak composition of n (cf [4]) if ∞ X

αi = n

i=1

1

where, αi ≥ 0. Clearly, only finitely many of the αi s can be non-zero, however the length of α is allowed to be infinite. Given a denumerable collection of indeterminates x = (x1 , x2 , . . .) and a weak composition α of n, we then define the monomial xα as:

xα =

∞ Y

i xα i

i=1

The above product is finite as only finitely many of the elements of α are allowed to be nonzero. Following [1] and [4] we begin by defining the ring of homogeneous symmetric functions of degree n. Let R be a commutative ring with identity and let n ∈ N. A symmetric function f (x) = f (x1 , x2 , . . .) is a formal power series:

f (x1 , x2 , x3 , . . .) =

X

cα x α

α

such that: 1. α ranges over weak compositions of n, 2. cα ∈ R, and 3. f (xσ(1) , xσ(2) , . . .) = f (x1 , x2 , . . .) for any permutation σ of the positive integers. Then, let ΛnR be the collection of all such formal power series in the indeterminates x1 , x2 , x3 , . . ., of degree n. It is clear that ΛnR is an R-module. We will write Λn when R is clear from the context. Note that Λ0R = R. We then define: Λ = ΛR =

∞ M

Λn

n=0

to be the ring of symmetric functions over the ring R. It is clear from the definition that Λ can be endowed with the structure of a graded algebra over R. In the following discussion, we will take R = Z unless otherwise noted. There are many remarkable bases for ΛZ and ΛQ , and we mention a few of them here and refer the reader to [4] for the full details. The problem of studying the transition matrices from one basis to another of Λ is interesting in its own right. We will be concerned mainly with the basis of Schur functions, which will be the focus of the next section. We begin with the monomial symmetric functions.

2

Definition 3.1. Let λ = (λ1 ≥ λ2 ≥ . . . ) be a partition of n. Then, X

mλ (x) =

xµ

µ: µ=σλ

where σ is some permutation of the positive integers, σλ = (λσ(1) , λσ(2) , . . .) and the sum is over all such distinct µ. The following result is well known: Theorem 3.1. (c.f. [4]) { mλ (x) | λ ∈ P ar } forms a basis of Λ as a module over Z. Likewise, we can also define the elementary symmetric functions (eλ ) and the complete homogeneous symmetric functions (hλ ): 1. Let en = m(1n ) , then eλ = eλ1 eλ2 . . ., 2. Let hn =

P

mλ , then hλ = hλ1 hλ2 . . .

|λ|=n

It is also well known that { hλ | λ ∈ P ar } and { eλ | λ ∈ P ar } are both basis of Λ as a Z-module. We refer the interested reader to [4]. Finally, there is an inner product that can be placed on Λ defined by the relation: hhλ , mµ i = δλ,µ where δλ,µ is the Kronecker delta. This inner product is known as the Hall inner product on Λ. The defining relation for the inner product implies that hsλ , sµ i = δλ,µ where sλ is the Schur function indexed by the partition λ. In particular, this means that the Schur functions introduced in the next section form an orthonormal basis of Λ.

3.2

Schur Functions

We can distinguish an important subset of Λ, called the Schur functions, indexed by partitions of n ∈ N. The Schur functions will also form a basis of Λ, however it is not obvious from the definition that the Schur functions are even symmetric. First we will need the notion of a semi-standard Young tableau (SSYT). Definition 3.2. Let λ ` n such that λ = (λ1 , λ2 , . . . , λk ). Then a Young diagram of shape λ is a collection of left justified boxes such that there are λ1 boxes in the first row, λ2 boxes in the second row, and so on to λk boxes in the kth row. We illustrate the above definition with the following example. 3

Example 3.1. Let λ = (5, 5, 2, 1, 1). Then |λ| = 14 and the corresponding Young diagram is:

Given a Young diagram of shape λ, a filling of the shape is an assignment of numbers to boxes of the diagram. There are particular types of fillings that we need to identify. Definition 3.3. Let λ be a partition. Then a semi-standard Young tableau (SSYT) is a filling of the shape λ with positive integers such that reading across rows form weakly increasing sequences and reading down columns form strictly increasing sequences. Example 3.2. Let λ be as in example 3.1. Then the following is a SSYT of shape λ: 3

7

4

9 10 10 11

8

8

8

5 15 8 9 We will next need the notion of a skew SSYT (or skew tableau). The following definition will be necessary towards this end. Definition 3.4. Let λ = (λ1 , λ2 , . . .) and µ = (µ1 , µ2 , . . .) be partitions. Then µ ⊆ λ if and only if µi ≤ λi for all i. The above definition makes precise the idea of one Young diagram being contained in another. If µ ⊆ λ, then we can remove µ from λ to produce a skew diagram. Definition 3.5. Let µ and λ be partitions such that µ ⊆ λ, then λ/µ is defined to be the shape obtained from λ by removing the boxes of µ. Example 3.3. Let λ = (5, 4, 1, 1, 1) and µ = (3, 2, 1) then µ ⊆ λ and the resulting skew shape λ/µ is given by the following diagram.

4

where the gray colored boxes indicate that they have been removed from the overall diagram. Given a skew shape λ/µ we can define a semi-standard Young tableau of skew shape λ/µ in a way analogous to the above. Definition 3.6. Let λ/µ be a skew shape. Then a semi-standard Young tableau of shape λ/µ is a filling of the boxes of λ/µ with positive integers such that every row forms a weakly increasing sequence and every column forms a strictly increasing sequence. Example 3.4. One example of a SSYT of the shape given in 3.3 is 5 3

5

7

6 7 Associated to any skew SSYT of shape λ/µ is the skew Schur function sλ/µ . Let SSYT(λ/µ) denote the set of all semi-standard Young tableaux of shape λ/µ (therefore, SSYT(λ/µ) is an infinite set). Definition 3.7. Let λ, µ ∈ P ar such that µ ⊆ λ and T ∈ SSY T (λ/µ). Define xT = xa1 1 xa2 2 . . . where ai is the number of times the number i appears in T . Then the skew Schur function is given by:

X

sλ/µ (x1 , x2 , . . .) =

xT

T ∈SSY T (λ/µ)

In the above definition, if µ = ∅, the empty partition, then sλ/µ is known as the Schur function indexed by λ. Example 3.5. Consider the partition λ = (3, 1, 1) and consider fillings of the shape λ with the numbers {1, 2, 3}. Then the corresponding segment of the Schur function s(3,1,1) is given by:

x31 x2 x3 + x1 x22 x23 + x21 x22 x3 + x21 x2 x23 + x1 x32 x3 + x1 x2 x33 and the corresponding SSYT are: 1

1

1

1

1

2

1

1

3

1

2

2

1

3

3

1

2

2

2

2

2

2

3

3

3

3

3

3

2

3

As noted above, it is not immediate from the definition that sλ/µ is a symmetric function , but it turns out that this is the case.

5

Theorem 3.2. (c.f. [4]) Let λ, µ ∈ P ar such that µ ⊆ λ, then sλ/µ ∈ Λ. In fact, the Schur functions (with µ = ∅) form a basis of Λ as a Z-module. Theorem 3.3. (c.f. [4]) {sλ | λ ∈ P ar } is a basis of Λ as a Z-module. Now that we have the fact that the Schur functions form a basis of Λ, we can state the following definition of the Littlewood-Richardson coefficients: Definition 3.8. The Littlewood-Richardson coefficients cνλ,µ are defined in the following equivalent ways. Let λ and µ be partitions such that µ ⊆ λ, then:

sν/λ =

X

cνλ,µ sµ

µ⊆ν

sλ sµ =

X

cνλ,µ sν

ν⊇µ,λ

where |ν| = |λ| + |µ|. It is a remarkable fact that cνλ,µ ∈ Z≥0 . Given that the Littlewood-Richardson coefficients are always non-negative we then state the famous Littlewood-Richardson problem: Problem 3.1. Find a combinatorial interpretation for the numbers cνλ,µ . Indeed, many combinatorial interpretations of the Littlewood-Richardson coefficients have been found. We delay these descriptions until chapter 3 after surveying some applications to representation theory and Schubert calculus.

6

Chapter 4

Applications to Representation Theory and Schubert Calculus The Littlewood-Richardson coefficients defined in the last chapter turn out to have many important applications to representation theory and Schubert calculus. We will survey some of these results here; omitting many of the details. These applications arise from the fact that the Schur polynomials play a major role in each of the applications we look at. In the sequel we will need the representation ring of a group G. See [2] for more details. Definition 4.1. The representation ring of a group G is the ring is given by:

h[V ] | V a representation of Gi / h[V ] + [W ] − [V ⊕W ]i where the multiplication is given by the tensor product of representations [W ][V ] = [W ⊗ V ] and extended linearly.

4.1

The Representation Theory of Sn

Let n be a positive integer. We denote by Sn the symmetric group on n letters. It is known that the number of irreducible representations of any finite group is the same as the number of conjugacy classes of that group. In Sn , each conjugacy class consists of permutations with the same cycle type, and each cycle type forms a partition of n. Therefore, the irreducible representations of Sn are indexed by λ ∈ P ar(n). Call these irreducible representations S λ .

7

The application of Schur polynomials is due to the Frobenius characteristic. Let Rn = R(Sn ) denote the representation ring of Sn . Note, if S λ is the irreducible representation of Sn indexed by the partition λ ` n,

 then, Rn = [S λ ] | λ ∈ P ar(n) . Then:

R=

∞ M

Rn

n=0

The Frobenius characteristic is a map ϕ : R → Λ such that the usual product of symmetric functions in Λ corresponds to induction on R. Formally, we begin as follows. Definition 4.2. Let R be as above. Define a product Rn × Rm → Rnm as:

i h S V  W [V ] ◦ [W ] = IndSn+m n ×Sm where  denotes the external tensor product between V and W , i.e. the standard tensor product with the action (σ, τ )(v  w) = σv  τ w. Then, the Frobenius characteristic ϕ has the following properties. Theorem 4.1. (c.f. [1]) Let ϕ : Λ → R be the Frobenius characteristic. Then, ϕ is an isometric isomorphism of algebras (with respect to the Hall inner product on Λ and the standard inner product of characters on R). Further, ϕ(sλ ) = [S λ ]. Which then yields the following corollary. Corollary 4.1. X

[S λ ] ◦ [S µ ] =

cνλ,µ [S ν ]

ν ` n+m λ, µ ⊆ ν

Proof. Let ψ = ϕ−1 : R → Λ. Then, since ϕ is an isomorphism, so is ψ. This then gives, ψ([S λ ] ◦ [S µ ]) = P ν ψ([S λ ])ψ([S µ ]) = sλ sµ . But, sλ sµ = cλ,µ sν . ν

Thus, computing the product in (R, ◦) is equivalent to the multiplication of Schur functions in Λ and then expanding the result in the basis of Schur functions. The coefficients of this expansion are exactly the Littlewood-Richardson coefficients by definition.

4.2

Representation Theory of GLn (C)

It is known that the irreducible polynomial representations of GLn (C) = GLn are indexed by partitions λ such that λ contains at most n parts. That is λ ` n and λ = (λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0). Denote the 8

irreducible representation of GLn indexed by λ be V λ .

 Let R = [V λ ] | λ ` n be the representation ring of GLn . Then the product defined on R is given by: [V ] · [W ] = [V ⊗ W ], i.e. the tensor product of irreducible representations. Let Λ(x1 , x2 , . . . , xn ) be the ring of symmetric functions restricted to the variables x1 , x2 , . . . xn . Then, define the character map: ch : R → Λ(x1 , x2 , . . . xn ). We then have the following fact. Theorem 4.2. The map ch : R → Λ(x1 , x2 , . . . , xn ) is an homomorphism of algebras: ch(v ⊗ w) = ch(v)ch(w) and [V λ ] 7→ sλ (x1 , x2 , . . . , xn ). This then yields the following corollary. Corollary 4.2. Vλ⊗Vµ =

M

ν

(V ν )⊕ cλ,µ

ν

That is, the Littlewood-Richardson coefficients give the multiplicities of the irreducible representation V ν appearing in the decomposition of the tensor product of two irreducible representations of GLn .

4.3

Schubert Calculus on the Grassmannian

Let E = Cn+r , then the Grassmannian, Grr (E), is defined as the collection of all r dimensional subspaces of E. Grr (E) can be realized as the quotient GLn+r (C)/P where P is the subgroup of non-singular matrices of the form: r

n+r





∗  n+r 0 r

∗ ∗

 

Thus, Grr (E) can be given a topology through this identification. We will see in a later section that the Grassmannian is an example of a partial flag variety and the corresponding subgroup P is a maximal parabolic subgroup. For each partition λ (or equivalently, Young diagram of shape λ) such that λ ⊆ [r]×[n], the r×n rectangle, we define the corresponding Schubert variety. Definition 4.3. Let e1 , e2 , . . . en+r be the standard basis vectors of Cn+r . Let

F• = ({0} ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fn+r = Cn+r ) be the standard flag with Fi = he1 , e2 , . . . , ei i. Then, the Schubert varieties are given by:

9

Ωλ (F• ) = Ωλ =



V ∈ Grr (Cn+r ) | dim(V ∩ Fn+i−λi ) ≥ i, 1 ≤ i ≤ r



where λ is a partition that fits in the r × n rectangle. In the above definition, we have chosen a particular reference flag for which to define the Schubert varieties, i.e. F• . It turns out that this choice does not lead to any loss of generality since Schubert varieties indexed by the partition λ defined with a choice of reference flag are related by a continuous action of GLn+r (C). Thus, the class induced by the Schubert variety in cohomology will be the same regardless of the reference flag used. We have the following important facts regarding Schubert varieties (in the following dimension always refers to dimension over C): 1. Ωλ =

` µ⊇λ

Ω◦µ = Ω◦λ where Ω◦λ = {V ∈ Grr (Cn+r ) | dim(V ∩ Fj ) = i for n + i − λi ≤ j ≤ n + i + 1 − λi+1 }

is the open Schubert cell indexed by the partition λ. That is, Ωλ is the closure of Ω◦λ . Observe that: a

Grr (Cn+r ) =

Ω◦λ

λ⊆[r]×[n]

Furthermore: dim Ω◦λ = nr − |λ|. 2. Grr (Cn+r ) is a smooth manifold such that dim Grr (Cn+r ) = nr and codim Ωλ = |λ| we have that Ωλ determines a class σλ = [Ωλ ] ∈ H 2|λ| (Grr (Cn+r )). 3. The set {σλ |λ ⊆ [r] × [n]} forms an additive basis for the cohomology ring H ∗ (Grr (Cn+r )). This can S be seen by setting Xi = |λ|=i Ωλ . Then X0 = Ω∅ = Grr (Cn+r ), Xnr = Ω(nr ) = he1 , . . . , er i, and we have the filtration: Grr (Cn+r ) = X0 ⊇ X1 ⊇ · · · ⊇ Xnr with Xi \ Xi+1 =

a

Ω◦λ

|λ|=i

then by applying fact (1) above and results from algebraic topology, we get that the Schubert classes form a basis of H ∗ (Grr (Cn+r )). In other words, the Schubert varieties give the Grassmannian the structure of a CW-Complex.

10

The product of Schubert classes can be given a geometric interpretation. First, let F˜• be the flag such that F˜i is the span of the last i basis vectors. Then, for partitions λ, µ ⊆ [r]×[n], we can define the Schubert ˜ µ = Ωµ (F˜• ). Despite the choice of flags, the induced Schubert classes in varieties: Ωλ = Ωλ (F• ) and Ω cohomology, σλ and σµ , are the same. The use the opposite flag F˜• for the Schubert variety indexed by µ is ˜ µ is proper and transverse. Then the product σλ σµ can be calculated by analyzing to guarantee that Ωλ ∩ Ω ˜ µ . For example we have the following duality theorem: the intersection Ωλ ∩ Ω Theorem 4.3. Suppose λ, µ are partitions such that: λ, µ ⊆ [r]×[n] and |λ| + |µ| = nr, then:

σλ σµ =

   1·[pt]

for µ = λ∨

  0

otherwise.

where [pt] indicates the class of a point and λ∨ is the compliment partition to λ in the r×n rectangle. By using this basis of Schubert classes, we can determine the structure of H ∗ (Grr (Cn+r )). The details are beyond the scope of this section, but the idea is as follows. Let Λ be the ring of symmetric functions as defined above, define the map φ : Λ → H ∗ (Grr (Cn+r )) given by sending the Schur polynomial sλ to its corresponding Schubert class σλ if λ ⊆ [r]×[n], otherwise sλ is mapped to 0. It follows that φ is an additive, surjective homomorphism. The more important result is, however, that φ is also a homomorphism of rings. This fact requires quite a bit of work to show. Since φ is a ring homomorphism, we can apply the corresponding isomorphism theorems to deduce that the cohomology ring of the Grassmannian is isomorphic to a quotient of the ring of symmetric polynomials in at most r indeterminates. We summarize this result in the following theorem. Theorem 4.4. Let E = Cn+r , then H ∗ (Grr (E)) ∼ = Sym(x1 , . . . , xr )/I for some ideal I. Furthermore, we have the following Littlewood-Richardson rule: Theorem 4.5. Let σµ and σλ be Schubert classes. Then:

σλ σµ =

X

cνλ,µ σν

|µ|+|λ|=|ν| [r]×[n] ⊃ ν ⊇ λ,µ

where cνλ,µ are exactly the Littlewood-Richardson coefficients described in the previous sections. Theorem 4.5 together with theorem 4.3 leads to a geometric interpretation of the Littlewood-Richardson coefficients.

11

  σλ σµ σν ∨ =  

 X

|µ|+|λ|=|τ | τ ⊇λ,µ

 cτλ,µ στ   σν ∨

= cνλ,µ ·[pt] by duality

Let E• be a flag in general position relative to F• and F˜• , then the product σλ σµ σν ∨ corresponds to the ˜ µ ∩ Ω(E• )ν ∨ . Thus, cν is the number of points in the triple intersection of these triple intersection Ωλ ∩ Ω λ,µ Schubert varieties.

12

Chapter 5

Jeu de Taquin and the Littlewood-Richardson Rule We now survey an answer to the question posed in chapter 2 and describe a combinatorial interpretation of the Littlewood-Richardson coefficients cλµ,ν . Our exposition is based on that found in an appendix of [4].

5.1

Jeu de Taquin

Let λ and µ be partitions such that µ ⊆ λ. We can then form the skew diagram λ/µ and consider standard Young tableaux of shape λ/µ. A standard Young tableau is a filling of the shape λ/µ from the set [n] such that each row and column form strictly increasing sequences. Jeu de taquin is essentially a game where each box of the given skew tableau is a game piece that can be moved based on given rules. We describe the process in the following definition and then consider an example. Definition 5.1. Let T be a standard Young tableau of shape λ/µ. Let b denote an empty box that can be added to T so that the resulting shape is a valid skew Young diagram with b’s bottom or right edge adjacent to a box of T . A jeu de taquin slide into b is based on the following rules: 1. If b has one neighbor, then slide the entry in the neighboring box into b creating a new empty box. 2. If b has two neighbors, then slide the smallest entry into b creating a new empty box. The above rules are repeated until the empty box is effectively ejected from the tableau upon reaching the outside border. Let jdtb (T ) denote the resulting tableau. 13

Example 5.1. Let λ = (6, 4, 2, 1) and µ = (3, 2). Let T1 be the following standard tableau of shape λ/µ.

2

b

3

1

6

5

8

4

7 The boxes of µ appear grey in the above diagram. Then, then result of jdtb (T1 ) is given by the sequence: 1

3

5

8

6 2

1 →

4

7

3

5

8

1

6 2

5

8

6

=

4

3

2

7

4

7

The tableau property of T is preserved at each move of the game by definition of the sliding rules, and therefore the resulting shape jdtb (T ) is again a tableau (when the empty box is ignored). Given a standard Young tableau of skew shape λ/µ, the jeu de taquin procedure can be repeated on all of the boxes of µ in order to produce a straight shape tableau. Example 5.2. Continuing with the skew shape defined in 5.1: 1 3 5 8 6 2 4 7

→

→

1 3 5 8 2 4 6 7

→

1 3 5 2 4 6 7

→ ··· →

1 3 5 8 4 6 2 7

=

1 3 5 8 4 6

→

2 7

7

→

1 3 5 8 2 4 6 7

→

1 3 2 4 6 7

→

1 3 5 8 2 4 6 7

→

1 3 5 8 2 4 6 7

8

1 3 5 8 2 4 6

5 8

1 3 5 8 2 4 6 7

A remarkable fact of this straightening process is recorded below. Lemma 5.1. (c.f. [4])) Let T be a standard Young tableau of shape λ/µ. Then, there is a unique straight shape standard tableau, jdt(T ), produced from T by using the jeu de taquin on all boxes of µ. Let jdt(T ) denote the unique straight shape tableau described by lemma 5.1 for the skew tableau T . In particular, the above lemma implies that no matter which valid sequences of boxes of µ are chosen for jeu de taquin, the resulting straight shape tableau is the same. For instance, in the example 5.1 above, one could begin with the second box in the second row rather than the third box in the first row, however lemma 5.1 ensures we would have ended up with the same tableau.

14

5.2

The Littlewood-Richardson Rule

We now state the Littlewood-Richardson rule in the form relating to jeu de taquin. Theorem 5.1. (c.f. [4]) Let λ, µ be partitions such that µ ⊆ λ. Let ν be a partition such that |λ| = |µ| + |ν|. Let P be any fixed SYT of shape ν. Then cλµ,ν is the number of SYT ,T , of shape λ/µ such that jdt(T ) = P . In other words:

cλµ,ν = |{T ∈ SY T (λ/µ) | jdt(T ) = P }| Note that in the statement of the theorem, the tableau P can be chosen arbitrarily. We illustrate the result with an example. Example 5.3. Let λ = (6, 4, 2, 1), µ = (3, 2) and ν = (4, 3, 1). Then cλµ,ν = 3. Let P be the following tableau of shape ν: 1

3

5

2

4

6

8

7 Consider the following skew tableaux T1 , T2 , and T3 : T1 =

3 5 8 1 6 2 4 7

T2 =

3 5 8 4 6 1 7 2

T3 =

1 3 5 6 8 2 4 7

Then jdt(T1 ) = jdt(T2 ) = jdt(T3 ) = P .

5.3

Growth Diagrams

The sequence of skew tableaux obtained from an instance of jeu de taquin can be summarized into a growth diagram. The growth diagram is a powerful tool with numerous applications and generalizations. In fact, growth diagrams play an integral role in the proof of the Littlewood-Richardson rule stated in the previous section. Let λ and µ be partitions such that µ ⊆ λ and let T ∈ SY T (λ/µ). Then, T can be viewed as a chain of partitions starting from µ and ending with λ. Each partition in the chain is obtained from an immediate predecessor by adding a single box. Starting with µ, each box is adjoined to produce an immediate successor according to the filling of the tableau T , i.e. the order of addition of boxes is given by the entries of T . We illustrate this idea with an example. Example 5.4. Let λ = (6, 4, 2, 1) and µ = (3, 2). Let T1 be the standard tableau given in example 5.1. That is, T1 is given by the diagram: 15

3 1 2

5

8

6

4

7 where the gray boxes are the boxes corresponding to µ that have been removed from λ. According to this particular filling, the chain of partitions we get is: 3

⊆

µ=

1

⊆

1 2 4

⊆

2

3 5

⊆

1

2

3 5 1 6

⊆

⊆

2 4

⊆

1

3

⊆

1 2 4

3 5 1 6 2 4 7

3 5 8 1 6 2 4 7

=λ

The fillings are preserved in order to emphasize the order of the boxes added to build the chain, however, these are only considered as partitions. In particular, the above chain of partitions can be written as:

(3, 2) ⊆ (3, 3) ⊆ (3, 3, 1) ⊆ (4, 3, 1) ⊆ (4, 3, 2) ⊆ (5, 3, 2) ⊆ (5, 4, 2) ⊆ (5, 4, 2, 1) ⊆ (6, 4, 2, 1)

Given a chain of partitions ∅ ⊂ · · · ⊂ µ (called the rectification order) and µ ⊂ · · · ⊂ λ (with λ/µ the skew tableau to rectify), via the bijection between standard Young Tableaux and chains described above, we can form the jdt growth diagram corresponding to λ/µ by writing the chain ∅ ⊂ · · · ⊂ µ as the left edge from bottom to top, and the chain µ ⊂ · · · ⊂ λ as the top edge from left-to-right. The remainder of the diagram is completed according to the local rule defined below. Definition 5.2. (c.f. [4]) Consider the following fragment of a jdt growth diagram, where π, µ, and λ are given, and ν is to be determined: λ µ

ν π

Then, ν is determined by the following rules: 1. If µ is the unique shape such that π ⊂ µ and µ ⊂ λ, then ν = µ. 2. Otherwise, there is a unique shape differing from µ and this must be ν. 16

The following example illustrates the growth diagram for the sequence of skew shapes obtained for the tableau T1 of example 5.1. Example 5.5. The growth diagram corresponding to the the example 5.1 is given by:

32

33

331

431

432

532

542

5421

6421

22

32

321

421

422

522

532

5321

6321

21

31

311

411

421

521

531

5311

6311

2

3

31

41

42

52

53

531

631

1

2

21

31

32

42

43

431

531

0

1

11

21

22

32

33

331

431

where partitions have been represented by strings, e.g. the partition (4, 3, 1) is written as 431 and the empty partition ∅ is written as 0. Let λ and µ be as above. Let T ∈ SY T (λ/µ). Then, the growth diagram for T is equivalently given by writing the corresponding chains for each of the results of jdt. The chain for T is written on top, then below it is jdtb (T ) for some box b, and so on, until the bottom row corresponds to the chain for jdt(T ). Thus the bottom left corner of the diagram will correspond to the partition ∅, the upper-left corner corresponds to the partition µ, and the top-right corner corresponds to the partition λ. Observe that the chain π ⊆ µ ⊆ λ, corresponds to adding two boxes to π to produce λ. The intermediate µ corresponds to some way of adding the first box. If λ is obtained from π by adjoining two boxes in the same row or the same column, then µ = ν in the corresponding fragment of the growth diagram. Otherwise, λ is obtained from π by adjoining two boxes in different rows or columns of π, which can be done in two possible ways to produce two distinct intermediate partitions, and so in this case µ 6= ν.

17

Chapter 6

Beyond Type A Grassmannians The use of the jeu de taquin in computing the cup product in the cohomology ring of the Grassmannian Grr (Cn+r ) has motivated much work in extending such a combinatorial rule to other spaces. These spaces are analogues of the Grassmannian corresponding to P being some general parabolic subgroup. In particular, let G be a connected, simple, Lie group, then a partial flag variety is obtained from G by taking the quotient with an appropriate choice of parabolic subgroup P , G/P . In their remarkable papers Hugh Thomas and Alexander Yong [5, 6] give a generalization of the jeu de taquin to (co)minuscule flag varieties, G/P . This is then applied in a similar way to compute cup products in the cohomology rings of these spaces. The authors’ construction leads to diagrams of boxes that are not necessarily Young diagrams of partitions, however, fillings of such diagrams are used to define a Jeu de taquin rule in a similar way to that for standard Young tableaux. Therefore, the given diagrams of skew shape can be rectified in a similar manner as well. The desired coefficients are then found to be the number of such fillings that rectify to a certain straight shape filling. We present a survey of their work below and show how it specializes to the case described in the previous chapter. We assume the reader is familiar with the standard notation and results about root systems. A standard reference for the subject is [3]. We begin with an overview of the results in [5, 6] in general, and then focus on some concrete examples for root systems of particular types.

18

6.1

(Co)Minuscule Flag Varieties

Let Φ be a root system, and let Π ⊆ Φ be a system of positive roots. Let ∆ ⊆ Π be the (unique) simple system contained in Π. Let W be the Weyl group generated by the set of reflections corresponding to the simple system ∆. That is: W = h sα | α ∈ ∆ i where sα is the reflection sending α to −α and fixing all vectors in the hyperplane orthogonal to α. If ∆P ⊆ ∆, then the collection of reflections corresponding to the simple roots in ∆P generate a subgroup of W known as a parabolic subgroup. Denote this subgroup by WP . If |∆P | = |∆| − 1, then the corresponding parabolic subgroup is called a maximal parabolic subgroup. Let ∆P ⊆ ∆ be such a subset of simple roots. Then let β(P ) ∈ ∆ \ ∆P be the root removed from ∆ to obtain ∆P . Definition 6.1. Let β(P ) be as above. Then, G/P is cominuscule if and only if we have that for all α ∈ Π if β(P ) appears in the expansion of α as a linear combination of simple roots, then it does so with coefficient 1. We will need a few more definitions and results from the theory of root systems before we proceed with the definition of the minuscule flag varieties. Recall, ∆ forms a basis for the vector space spanned by the roots Φ such that the coefficients appearing in the expansion of any root are all of the same sign. Definition 6.2. Let α ∈ Φ, then the corresponding coroot is given by α∨ =

2α (α,α) .

Define Φ∨ := {α∨ | α ∈

Φ}. It is known that if Φ is a root system, then so is Φ∨ , called the dual system to Φ. Likewise, if ∆ ⊂ Φ is a simple system, then so is ∆∨ ⊂ Φ∨ . Therefore, any coroot can be expanded as a linear combination of the simple co-roots. Next, the concept of weights will play a fundamental role in the definition of the minuscule flag varieties, so we record the required definitions here. ˆ Definition 6.3. Let Φ be a root system and Φ∨ its dual. The weight lattice is given by L(Φ) = {λ ∈ ˆ ∨) = V | (λ, α∨ ) ∈ Z for all α ∈ Φ} where V = span(Φ). Likewise, the coweight lattice is given by L(Φ {λ∨ ∈ V | (λ∨ , α) ∈ Z for all α ∈ Φ} We use the pairing notation given by: hα, βi =

2(α,β) (β,β) .

Observe that hα, βi = (α, β ∨ ). We next define the

fundamental weights, which act as a dual basis to ∆ in a certain sense.

19

Definition 6.4. Let ∆ = {αi | 1 ≤ i ≤ k} be a simple system. Then, the fundamental weights {ωi | 1 ≤ i ≤ k} are defined by the relation: (ωi , αj∨ ) = (ωi∨ , αj ) = δi,j , where δi,j is the Kronecker delta. ˆ Observe, ωi ∈ L(Φ) moreover they generate the lattice; this can be seen by taking the simple root expansion of α ∈ Φ and applying definition 6.3. We can now state the definition of the minuscule flag varieties. Definition 6.5. Let β(P ) be as above. Then G/P is minuscule if and only if the associated fundamental weight ωβ(P ) satisfies: (ωβ(P ) , α∨ ) ≤ 1 for all α ∈ Π. The above definitions of (co)minuscule flag varieties deserve further elaboration. Let α ∈ Π, then we P can expand α as a linear combination of simple roots: α = αj ∈∆ cj αj . By the definition above of the ∨ fundamental weights, we get that (α, ωi∨ ) = ci . In the case of cominuscule flag varieties, by setting ωβ(P ) to ∨ be the fundamental weight of β(P ), we get that (α, ωβ(P ) ) = cβ(P ) ≤ 1 for all α ∈ Π. Additionally, we can P expand α∨ in the basis of simple co-roots ∆∨ : α∨ = α∨ ∈∆∨ ci αi∨ . The minuscule condition give us that i

∨

(α , ωβ(P ) ) = cβ(P ) ≤ 1.

6.2

A Jeu de Taquin Procedure for (Co)Minuscule Flag Varieties

Following [5], let ΩG = (Π, π(r + 1) < π(r + 2) < · · · < π(n). For example, in the case of k = 4 and n = 7, one such permutation (in one line notation) is given by: 2457136. Let w ∈ W, then let inv(w) = {α ∈ Π | w(α) ∈ −Π}. That is, inv(w) is the collection of positive roots sent to negative roots by the action of w. We next describe the bijection between W P and lower order ideals of ΛG/P . Lemma 6.1. The following diagram of bijections commutes: inv

YG/P S

WP

f

Pk×n−k where Pk×n−k is the collection of partitions contained in the n×n − k rectangle, inv : w ∈ W P 7→ inv(w) the set of inversions of w, S denotes superimposing the boxes of the partitions onto the corresponding roots in the lower order ideal, and f is to be described. Proof. The proof follows from the following facts: 1. If α ∈ Π then α = ei − ej for some i < j and its expansion as a linear combination of simple roots is given by: j−1 X

e` − e`+1

`=i

2. w ∈ W P ⇒ (k, k + 1) is an inversion of w and if α ∈ inv(w) then α = ei − ej such that: 1 ≤ i ≤ k < k + 1 ≤ j ≤ n. In particular, if α ∈ inv(w) then the (co)minuscule root must appear in its simple root expansion, i.e. α ∈ ΛG/P . 3. If (i, j) is an inversion of w ∈ W P then (i + 1, j), (i + 2, j), . . . , (k, j) are also inversions. Additionally, (i, j − 1), (i, j − 2), . . . , (i, k + 1) are also inversions. 24

Using 1, 2, 3 above, we get that inv(w) contains only roots in ΛG/P and forms a lower order ideal. The map f is described in [1]. Let w ∈ W P , then:

f (w) = λ = (w(k) − k, w(k − 1) − (k − 1), . . . , w(2) − 2, w(1) − 1)

which is clearly a bijection. Example 6.1. Let w = 2457136 then f (w) yields then the partition λ = (3, 2, 2, 1). Our goal is to generalize the above correspondences to other types. In the sections below, we show how analogues of the above bijections can be carried out in type Cn . Upon these bijections, SY T of the given type can be viewed as chains in Bruhat order and the jdt algorithm then corresponds to growth diagrams with a certain set of local rules.

6.4

Example: Root Systems of Type Bn

In this section we consider an example of type Bn . Recall that the Dynkin diagram for this type is given by the graph: ... 1

2

k

... n−1

n

with the black node corresponding to the only co-minuscule node. Let e1 , e2 , . . . , en again be the standard basis vectors of Rn . In this example, consider Φ = {±ei ±ej | i 6= j} ∪ {±ei | i = 1, 2, . . . n}. Then we set Π = {ei − ej | i < j} ∪ {ei | i = 1, 2, . . . , n} as our positive system, which gives us the simple system ∆ = {ei − ei+1 | i = 1, 2, . . . n} ∪ {en }. The Weyl group in this case is isomorphic to (Z2 )n o Sn . Let n = 5, in this case the co-minuscule node corresponds to the simple root e1 − e2 . The root lattice ΩG is given by:

25

e1 + e2 e1 + e3 e1 + e4 e1 + e5 e1

e1 − e3

e2

e1 − e2

e2 − e3

e3

e4 + e5 e4

e3 − e5

e3 − e4

e3 + e4

e3 + e5

e2 − e5

e2 − e4

e2 + e4

e2 + e5

e1 − e5 e1 − e4

e2 + e3

e4 − e5

e5

In the above poset, the nodes contained in a box correspond to the sub-poset ΛG/P , and the shaded boxes indicate an example of a lower order ideal contained in ΛG/P . Let λ be the straight shape given by the indicated ideal. The interpretation of such an ideal as a straight shape consisting of boxes corresponding to each of the elements in the ideal is given by the diagram:

where the shaded boxes indicate the elements of λ, and the white boxes correspond to the remaining elements of ΛG/P not included in the ideal.

6.5

Example: Root Systems of Type Cn

We now consider root systems of type Cn . The Dynkin diagram for this type is quite similar as for the case of Bn . The only differences are the cominuscule node and the shorter root. ... 1

2

... k n−1

n

In this case, the root system Φ = {±(ei ± ej ) | i ≤ i < j ≤ n} ∪ {±2ei | i = 1, . . . , n}. Then, the simple system is ∆ = {ei − ei+1 | i = 1, . . . , n − 1} ∪ {2en }. In the sequel, we identify the root ei + ej with the pair (i, ) and the root 2ei with the pair (i, ı). The Weyl group generated by the simple reflections in this case is the group of signed permutations, which we denote by Bn . Note that this is the same as in the case of root systems of type Bn above (hence 26

the overloaded notation). Elements of Bn are represented in window notation. First, we denote the element −i by ı for i ∈ [n]. We linearly order the set [n] = {n, n − 1, . . . , 1, 1, 2, . . . , n − 1, n} by:

1 < 2 < · · · < n < n < n − 1 < · · · < 1.

An element w ∈ Bn is thus viewed as a permutation of the set [n] such that w(ı) = w(i). The signed permutation w ∈ Bn is then completely determined by its window notation: w(1). . .w(n). Let P = ∆\{n}, and let WP denote the maximal parabolic subgroup generated by simple reflections given by the simple roots in P . Based on the Dynkin diagram above, we see that WP ∼ = Sn , the symmetric group on n letters. Thus, the parabolic quotient is Bn /Sn . Denote by W P the set of lowest coset representatives of Bn /Sn in the Bruhat ordering. It is clear that: W P = {w ∈ Bn | w(i) < w(i + 1) for 1 ≤ i ≤ n − 1}.

Intuitively, since we quotienting out by Sn , the first n letters of any w ∈ Bn can be “sorted” into increasing order while remaining in the same coset while decreasing its length as much as possible. It is a well known fact that we have a bijection: Bn /Sn ↔ W P . We first illustrate the root poset ΩG along with the sub-poset ΛG/P for C5 . Example 6.2. As in the above section, let n = 5, then the co-minuscule root is 2e5 , and the root poset, ΩG , is given by: 2e1 e1 + e2 e1 + e3 e1 + e4 e1 + e5 e1 − e5 e1 − e4 e1 − e3 e1 − e2

e2 − e5

e2 − e4

e2 − e3

27

2e3

e3 + e4

e3 + e5

e3 − e5

e3 − e4

e2 + e3

e2 + e4

e2 + e5

2e2

2e4

e4 + e5

e4 − e5

2e5

where the boxed nodes are those in ΛG/P , and the shaded nodes give an example of a lower order ideal in this poset. The lower order ideal then translates to the following shape in the staircase diagram:

We continue with an example run of the jdt algorithm using the lower order ideal given above. Example 6.3. Let λ ∈ YG/P be the lower-order ideal considered above. Let µ ⊂ λ be the lower-order ideal of ΛG/P consisting of the roots {e4 + e5 , 2e4 , 2e5 }, that is, µ is given by the diagram:

where the remaining boxes of the poset ΛG/P are included and the boxes of µ are shaded. Consider the skew shape λ/µ with diagram:

where the above diagram is shown without the remaining boxes of ΛG/P . Then, to complete the setup, consider the filling T ∈ SY TG/P (λ/µ):

4

5

1

2

3

We next proceed to illustrate the jdt slides leading to the rectification of the filling. 4 5 1 2 3 x

−→

4 5 1 x 3 2

−→

4 5 1 3 x 2

= jdtx (T ) = T 0

4 5 1 3 y 2

−→

4 5 y 3 1 2

−→

4 5 3 y 1 2

−→

4 3 5 1 2 z

−→

4 3 5 z 2 1

−→

4 3 5 2 z 1

−→

4 y 3 5 1 2

4 3 z 2 5 1

Note that the final shape, rect(T ) is a straight shape. 28

= jdty (T 0 ) = T 00

= jdtz (T 00 ) = rect(T )

Example 6.4. Let λ ∈ YG/P be the above ideal. Consider the lower order ideal ν ⊂ λ given by:

which corresponds to the roots e3 + e5 , e4 + e5 , and 2e5 . The skew shape λ/ν is then:

where the diagram is shown without the ambient boxes of the remaining elements of ΛG/P . Let T be the filling of the above shape giving by:

4

5 2

3

1

where the filled boxes are those of the skew shape λ/ν and the empty boxes are those of ΛG/P \ λ/ν. Notice that there are two empty boxes in the above diagram, corresponding to the roots e3 + e5 and e4 + e5 , that are not contained in the shape λ/ν and are covered by elements of the shape λ/ν. The rectification of T , rect(T ), is then given by the following sequence of jdt slides. Note that rect(T ) is independent of the sequences of boxes for jdt slides. Therefore, we may choose either one of the two boxes mentioned in the above paragraph to begin with. Let x correspond to the box given by the root e3 + e5 then jdtx (T ) is given by the sequence:

4 5 x 2 3 1

−→

4 5 2 x 3 1

−→

4 5 2 3 x 1

=

4 5 2 3 1

We then continue on with the remaining jdt slides (where the lettered box is the one chosen as for the jdt slide): 4 5 2 3 y 1

−→

−→

4 z 5 2 3 1

4 5 2 3 1 y

−→

−→

z 4 5 2 3 1

4 5 2 y 1 3

=

−→

4 y 2 5 1 3

=

4 5 2 3 1

This result is a straight shape standard tableau. 29

4 2 5 1 3

−→

4 2 5 1 3 z

−→

4 2 5 z 3 1

We next show how to extend the results for root systems of type An−1 to root systems of type Cn . Our goal is to translate standard Young tableaux to chains in Bruhat order thus allowing us to translate the jdt procedure to growth diagrams. This is effectively realized upon the bijections give in the lemma below. Lemma 6.2. We have the following bijections: YG/P

inv

S

WP

f

PG/P where the set PG/P is a collection of pairs, (h, τ ) where 0 ≤ h ≤ n and τ is a partition contained in the (n − h) × h rectangle. Proof. We first illustrate the bijection f : W P → PG/P . Let w ∈ W P . Then, as noted above: w(i) < w(i+ 1) for 1 ≤ i ≤ n − 1. Let k be the index of the last positive symbol of w in window notation. Then, set τ = (w(k) − k, w(k − 1) − (k − 1), . . . , w(1) − 1) and let h = |{i | w(i) < 0 for 1 ≤ i ≤ n}|. Clearly f is a bijection as the inverse function f −1 simply reverses the process of constructing τ which, along with h, uniquely determines an element of W P . Let S : PG/P → YG/P be the bijection (τ , h) 7→ λ ∈ YG/P given by superimposing the boxes of the Young diagram for τ onto the poset ΛG/P at height given by h. The remaining bijection is given by the map inv : W P → YG/P where inv(w) is the set of inversions of w. Since w ∈ W P , all of its inversions are of the form (i, ) for i ≤ j. Such an inversion corresponds to a root: ei + ej in the case i < j or 2ej in the case i = j. The collection of such roots for a particular w gives a lower-order ideal in ΛG/P . Using the above bijections we can construct growth diagrams consisting of Bruhat chains in Bn /Sn . Let λ, µ ∈ YG/P such that µ ⊂ λ. Let T ∈ SY T (λ/µ). Then, the left edge of the growth diagram corresponds to the chain given by the rectification order determined by µ, i.e. the order of the jdt slides into µ. The top edge is determined by λ corresponding to the order of adding boxes to µ to obtain λ determined by the filling T . The rest of the diagram is filled in corresponding to a local rule. We give a local rule for constructing these growth diagrams analogous to Fomin’s rule for the case of growth diagrams consisting of chains of partitions, i.e. the Type An−1 case. In the growth diagram, since the left and top chains are determined, the following local rule is applied to fill in the rest of the diagram starting from the top left corner. Definition 6.9. Consider the following fragment of the jdt growth diagram where w, u, and v are given, and x is to be determined: 30

u

v

w

x

In W it is known that the structure of the interval [w, v] in Bruhat order is a diamond: β

v

δ u0

u α

w

γ

then, x is determined by the following local rule: 1. If u0 ∈ W P then set x = u0 , otherwise 2. x = u. Observe, in the case of type Cn , the rule becomes much simpler due to the condition that the elements of ΛG/P are of the form ei + ej for i ≤ j: If (α, β) = 0 then x = wsβ , γ = β, and δ = α, otherwise x = u. Note that (α, β) = 0 ⇔ sα sβ = sβ sα , i.e. the reflections commute. We illustrate the local by continuing with the above examples. First, we give the growth diagram corresponding to example 6.3. 123 5 4

124 5 3

125 4 3

12 5 4 3

13 5 4 2

14 5 3 2

12354

12453

124 5 3

125 4 3

135 4 2

145 3 2

12345

1235 4

123 5 4

124 5 3

134 5 2

135 4 2

12345

12345

12354

1245 3

1345 2

134 5 2

The growth diagram for example 6.4 is given by:

31

12453

124 5 3

125 4 3

12 5 4 3

13 5 4 2

14 5 3 2

12354

123 5 4

124 5 3

125 4 3

135 4 2

145 3 2

12345

1235 4

1245 3

124 5 3

134 5 2

135 4 2

12345

12345

12354

123 5 4

124 5 3

125 4 3

Each node of the right chain does indeed correspond to the lowest coset representative given by the result of each jdt slide of the example above. The bottom right node then gives the element of w ∈ W P determined by the rectification of T .

6.6

Example: Root Systems of Type Dn

We briefly describe an example in the root system of type Dn . In this case, the Dynkin diagram is given by: n ... 1

2

3 n−1

where the black node indicates the choice of (co)minuscule node. Note that nodes n and n − 1 can also be chosen, but the poset ΛG/P corresponding to those choices are isomorphic to that of type Cn−1 . Note further that, in type Dn , cominuscule and minuscule coincide. In this case, Φ = {±(ei ± ej ) | 1 ≤ i < j ≤ n} and the simple system is given by ∆ = {ei − ej | i = 1, . . . , n − 1} ∪ {en−1 + en }. We once again consider the case when n = 5. The poset ΩG , however becomes much harder to draw, and so we adopt a different notation from the above cases of types An−1 , Bn , and Cn in order to simplify the presentation. In the Hasse diagram below, a label of the form (i, j) represents a root of the form ei − ej and a label of the form (i, ) indicates a root of the form ei + ej . Nodes are included to further clarify the structure of the post.

32

(1, 2)

(1, 3)

(1, 4)

(1, 5)

(2, 3)

(2, 4)

(1, 5)

(3, 4) (2, 5)

(2, 5) (1, 4)

(3, 5) (1, 3)

(1, 2)

(2, 4)

(3, 5)

(2, 3)

(3, 4)

(4, 5)

(4, 5)

The sub-poset ΛG/P then has the form: (1, 2) (1, 3) (1, 4)

(1, 5)

(1, 5)

(1, 4) (1, 3) (1, 2)

An example of a lower order ideal in the lattice YG/P is then given by selecting all nodes including and below the node (1, 4):

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Bibliography [1] William Fulton. Young tableaux. With applications to representation theory and geometry, volume 35 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1997. [2] William Fulton and Joe Harris. Representation theory. A first course, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. Readings in Mathematics. [3] James E. Humphreys. Reflection groups and Coxeter groups, volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1990. [4] Richard P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. [5] Hugh Thomas and Alexander Yong. A combinatorial rule for (co) minuscule schubert calculus. Advances in Mathematics, 222(2):596–620, 2009. [6] Hugh Thomas, Alexander Yong, et al. Cominuscule tableau combinatorics. In Schubert Calculus—Osaka 2012, pages 475–497. Mathematical Society of Japan, 2016.

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