Sign hibi cones and pieri algebras for the general linear groups

SIGN HIBI CONES AND PIERI ALGEBRAS FOR THE GENERAL LINEAR GROUPSWANG YI B.Sc., ECNU, China A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UN

SIGN HIBI CONES AND PIERI ALGEBRAS FOR THE GENERAL LINEAR GROUPS

WANG YI

(B.Sc., ECNU, China)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MATHEMATICS

NATIONAL UNIVERSITY OF SINGAPORE

2013

To my parents

v

First of all, I would like to thank my supervisor Professor Lee Soo Teck for hisguidance, encouragement and patience What I have learnt from him is not onlymathematics but also the way of life and the attitude to research I feel very proud

to be his student

I would like to thank Professor Roger Howe and Professor Sangjib Kim for theirinterest in my work Conversations with them enrich my knowledge and clear mydoubts

I would like to thank Ji Feng and Ma Jia Jun for many helpful discussions andsuggestions I learn a lot from them

I would like to thank Gao Rui, Li Xudong and Hou Likun for the two years happylife when we live together

I would like to thank Sun Xiang who helped me solving problems of LATEX

vii

I would like to thank Li Shangru and Yuan Zihong for memorable trips.

Lastly, I would like to thank my parents for their encouragement and support

2.1 Representations of Linear Algebraic Groups 5

2.2 Rational Representations of GLn 8

2.3 Generalized Iterated Pieri Rules for GLn 12

2.3.1 Polynomial Iterated Pieri Rule 12

2.3.2 Generalized Iterated Pieri Rule 15

2.4 Posets and Hibi Cones 21

2.4.1 Posets 21

2.4.2 Increasing Subsets and Decreasing Subsets 22

2.4.3 Hibi Cones 23

2.4.4 The Hibi Cone ZΓn,k,h , ≥0 26

2.5 Standard Monomial Theory for Hibi Algebras 30

2.5.1 Standard Monomial Theory 30

ix

2.5.2 Semigroup Algebras 31

2.5.3 Semigroup Algebras on Hibi Cones 31

2.6 (GLn, GLk) Duality 32

2.7 An Overview of Pieri Algebras 33

2.8 Polynomial Iterated Pieri Algebras 35

2.8.1 The Construction of the Polynomial Iterated Pieri Algebras 35 2.8.2 Polynomials Associated with Tableaux 37

2.8.3 Monomial Ordering and Sagbi Basis 39

2.8.4 The Structure of Rn,k,h 42

2.8.5 Reciprocity Algebras 44

2.8.6 General Iterated Pieri Algebras 45

3 Sign Hibi Cone 49 3.1 The Structure of the Sign Hibi Cone 49

3.2 Subsemigroups of Sign Hibi Cone 53

3.3 The semigroup Ωn,k,l 61

3.4 Further Results 67

3.4.1 Expression of Elements in a Sign Hibi Cone 67

3.4.2 Further Structure of ΩA,B 70

4 Anti-row Iterated Pieri Algebras 73 4.1 Construction 73

4.2 Generators of An,k,l 75

4.3 Leading Monomials 84

4.4 Structure of Anti-row Iterated Pieri Algebras 91

4.5 Applications to Howe Duality 93

4.6 Future Research 96

Contents xi

n , and write ρD,E

n as ρD

combinatorial description of how a tensor product of the form ρDn ⊗ ρE

to consider a more general version of the Pieri Rule, that is, a description of how

1

tensor products of the form

in-In [HKL], the authors studied the structure of the algebra An,k,p,h,l in the stablerange, that is, k+p+h+l ≤ n In this case, the structure of the algebra is controlled

by a semigroup H, called a Hibi cone ([Ho05]) The semigroup H has a very niceand simple structure: It has a finite set G of generators, and on which one can define

a partial ordering Each nonzero element f of H has a unique standard expression

On are obtained in the paper [KL]

This thesis has two objectives The first objective is to study the structure of a class

of semigroups which are more general than Hibi cones and are potentially useful in

representation theory A Hibi cone is constructed from a finite poset Γ; it is the set

ZΓ,≥0 of all order preserving functions f : Γ → Z≥0 with semigroup operation given

by the addition of functions It is natural to consider order preserving functions on

Γ which are allowed to take negative values Thus we consider the semigroup ZΓ,

of all order preserving functions f : Γ → Z, and we call it a sign Hibi cone More

generally, if A and B are two subsets of Γ, then we let

ΩA,B = {f ∈ ZΓ, : f (A) ≥ 0, f (B) ≤ 0}

If A = Γ and B = ∅, then ΩA,B = ZΓ,≥0 Thus ΩA,B is a more general construction

than a Hibi cone We show that ΩA,B retains many nice properties of a Hibi cone In

fact, it is generated by 2 subsemigroups which are Hibi cones, and it has a canonical

set of generators Moreover, each nonzero element of ΩA,B has a unique canonical

expression in terms of the generators

The second objective is to determine the structure of the anti-row iterated Pieri

algebra An,k,l = An,k,0,0,l without the stable range condition We show that a

semigroup Ωn,k,l is naturally associated with An,k,l, but Ωn,k,l is not a Hibi cone It

is a subsemigroup of a sign Hibi cone ZΓ n,l , and is of the form ΩA,B The semigroup

Ωn,k,l has a canonical set Gn,k,l of generators For each element in Gn,k,l, we associate

with it an element in the algebra An,k,l and let Gn,k,l be the set of elements of An,k,l

obtained in this way We show that the set Bn,k,l of standard monomials on Gn,k,l

forms a basis for An,k,l, and An,k,l has a flat deformation to the semigroup algebra

C[Ωn,k,l] on Ωn,k,l

Our results on the anti-row iterated Pieri algebras also have applications in Howe

duality We show that a subset of Bn,k,l can be identified with a basis for the

subspace of an irreducible lowest weight module of the general linear algebra glk+l

spanned by all the glk highest weight vectors

The outline of this thesis is as follows: In Chapter 2, we define some notations andconcepts which are needed in the thesis We also discuss the generalized Pieri ruleand summarize several existing types of Pieri algebras In particular, we review insome details the structure of polynomial iterated Pieri algebras In Chapter 3, westudy the structure of a sign Hibi cone and its subsemigroups Finally, we apply theresults of Chapter 3 to study the structure of the anit-row Pieri algebras in Chapter4.

Chapter 2

Preliminaries

In this chapter, we shall review some concepts and results which are related to thePieri algebras for the GLn

Let V be a complex vector space The set of invertible linear transformations from

V to itself forms a group under composition This group is denoted by GL(V ).Definition 2.1.1 ([GW]) Let G be a group

(a) A representation of G is a pair (ρ, V ), where V is a complex vector spaceand ρ : G → GL(V ) is a group homomorphism In this case, we also call V

sub-5

Definition 2.1.2 Let (ρ, V ) and (τ, W ) be two representations of a group G.(a) A complex linear map T from V to W is called an intertwining operator if

T ρ(g) = τ (g)T for all g ∈ G We shall denote by HomG(V, W ) the set of allintertwining operators T : V → W

(b) The representations V and W are called equivalent or isomorphic, denoted

by V ∼= W , if there exists an invertible operator in HomG(V, W )

If ρ : G → GL(V ) and σ : G → GL(U ) are representations, then there are standardways [FH] to define representations of G on the following spaces:

(a) The dual space V∗ of V

(b) The tensor product V ⊗ U

(c) The kth tensor power V⊗k

(d) The kth symmetric power SkV

(e) The kth exterior power Vk

V The representation of G on V∗ is called the contragredient or dual representation

of (ρ, V )

Next, let GLn := GLn(C) be the set of all invertible n × n complex matrices der matrix multiplication, GLn forms a group, called the complex general lineargroup We also let Mn = Mn(C) be the set of all n × n complex matrices, andlet P(Mn) be the algebra of complex polynomial functions on Mn We shall denotethe system of standard coordinates on Mn by (xij)1≤i,j≤n, so that P(Mn) can beregarded as a polynomial algebra on these variables

Un-2.1 Representations of Linear Algebraic Groups 7

Definition 2.1.3 ([GW])

(a) A subgroup G of GLn is called a linear algebraic group if there is a subset

S of P(Mn) such that

G = {g ∈ GLn: f (g) = 0 ∀f ∈ S}

(b) A regular function on GLn is a function φ : GLn → C such that φ can be

written as a polynomial on xij with 1 ≤ i, j ≤ n and det1 Here det is the

determinant function on GLn

(c) If G ⊆ GLnis a linear algebraic group, then a function on G is called regular

if it is the restriction of a regular function on GLn to G

Definition 2.1.4 ([GW]) Let ρ : G → GL(V ) be a representation

(a) ρ is called rational (or regular) if

1) V is finite dimensional, and

2) for all λ ∈ V∗ and v ∈ V , the function

φλ,v(g) = λ (ρ(g)v)

is regular

(b) ρ is called locally regular if

1) V has countable dimension, and

2) every finite-dimensional subspace U of V is contained in a finite-dimensionalG-invariant subspace W such that the restriction of ρ to W is a rational

representation

We shall denote by ˆGr the set of all equivalent classes of irreducible rational

repre-sentations of G

Theorem 2.1.1 ([GW]) Let G be a linear algebraic group and let ρ : G → GL(V )

be a locally regular representation Then V has a primary decomposition given by

multi-mV(U ) := dim HomG(U, V )

If mV(U ) < ∞ for all U ∈ bGr, then we shall also write

U ∈ ˆ G r

mV(U )U

We now briefly review the highest weight theory for GLn ([GW], [Hum]) Let Bn

be the subgroup of GLn consisting of all upper triangular n × n invertible complexmatrices Then Bn is a Borel subgroup of GLn Let An be the subgroup of GLnconsisting of all diagonal matrices in GLn and let Un be the subgroup of GLn con-sisting of all upper triangular matrices with 1’s on the diagonal Then Bn = AnUn

For each α = (α1, , αn) ∈ Zn, let ψαn : An → C be defined by

2.2 Rational Representations of GLn 9

Definition 2.2.1 Let (ρ, V ) be a rational representation of GLn

(a) A nonzero vector v of V is called a weight vector if there exists α ∈ Zn such

that

ρ(a)(v) = ψnα(a)vfor all a ∈ An In this case, we say that v has weight ψα

n.(b) A weight vector v of V is called a highest weight vector if

ρ(u)(v) = vfor all u ∈ Un

We now let

Λ+n := {λ = (λ1, , λn) ∈ Zn|λ1 ≥ ≥ λn} (2.2)Theorem 2.2.1 ([GW]) Let ρ : GLn→ GL(V ) be an irreducible rational represen-

tation

(a) V has a highest weight vector v which is unique up to scalar multiple

(b) The weight of v is of the form ψλ

n for some λ ∈ Λ+

n.(c) Up to equivalence, the representation ρ is completely determined by λ Thus

we say that ρ has highest weight ψnλ and denote it by ρλn

Next, we let

Λ++n := {λ = (λ1, , λn) ∈ Λ+n|λn ≥ 0} (2.4)

We call the representations ρλn (λ ∈ Λ++n ) polynomial representations Theserepresentations can be indexed by certain Young diagrams

Definition 2.2.2 ([Fu]) A Young diagram, or Ferrers diagram, is a collection

of square boxes arranged in left-justified rows with each row no longer than the oneabove it

We usually denote a Young diagram by a capital letter such as D, E, F etc If D is

a Young diagram with at most n rows and there are dj boxes in the jth row, then

we shall write

D = (d1, , dn)

Note that we allow dj = 0 So two sequences (d1, , dn) and (d01, , d0m) with

n > m represent the same Young diagram if and only if ds= d0s for 1 ≤ s ≤ m and

dt = 0 for t > m If the kth row is the last non-zero row of D, we will say that Dhas depth k and write depth(D) = k We also denote by |D| the totaly number ofboxes in D

We shall identify Λ++n with the set of all Young diagrams with depth less than orequal to n If λ ∈ Λ++

n corresponds to the Young diagram D, then we shall alsowrite ψλn and ρλn as ψDn and ρDn respectively

Example 2.2.1 The Young diagram

is denoted by D = (5, 4, 2, 2, 1) The depth of D is 5 and |D| = 14

2.2 Rational Representations of GLn 11

Example 2.2.2 For positive integer α, the Young diagram consisting of only one

row and α boxes is denoted by (α) Then ρ(α)n ∼= Sα(Cn) In particular, ρ(1)n ∼= Cn is

the standard representation of GLn

Example 2.2.3 For a positive integer β, let 1β denote the Young diagram with

only one column and β boxes That is,

Example 2.2.4 For each positive integer m, let detm : GLn → C be defined by

detm(g) = (det g)m for all g ∈ GLn

Then detm can be regarded as an one-dimensional rational representation of GLn,

λ− = (min{λ1, 0}, , min{λn, 0})

Then λ+, (λ−)∗ ∈ Λ++

n , so that each of λ+ and (λ−)∗ can be identified with aYoung diagram Let D and E be the Young diagrams corresponding to λ+ and λ−∗respectively Then one can check that depth(D) + depth(E) ≤ n We shall write

In this section, we shall review the Pieri rule for GLnand some of its generalizations

We first consider the problem of decomposing the tensor product of a polynomialrepresentation of GLn with representations indexed by one-row Young diagrams.Definition 2.3.1 If λ = (λ1, , λn), µ = (µ1, , µn) ∈ Λ+n and

µ1 ≥ λ1 ≥ µ2 ≥ λ2 ≥ µn ≥ λn,then we say λ interlaces µ and write λ v µ

Theorem 2.3.1 (The Pieri rule) Let D ∈ Λ++n and α ∈ Z≥0 Then

For a proof, see [GW] or [Ho95]

By iterating the Pieri rule, we obtain the following (see [HL12] and [HKL]):

Theorem 2.3.2 (Polynomial Iterated Pieri rule)

!

F

KF /D,αρFn,

2.3 Generalized Iterated Pieri Rules for GLn 13

where KF/D,α is the number of sequences (D0, D1, , Dh) of Young diagrams such

that

1) D0 = D, Dh = F ,

2) D0 v D1 v D2 v · · · v Dh and |Ds−1| + αs = |Ds| for all 1 ≤ s ≤ h

There are two other descriptions of the number KF /D,α The first one is related to

semistandard tableaux which we now explain

Definition 2.3.2 ([Fu])

(a) If D = (d1, , dn) and F = (f1, , fn) are Young diagrams, then we say that

D is contained in F and write D ⊆ F if ds≤ fs for all 1 ≤ s ≤ n

(b) If D ⊆ F , then by removing the boxes of D from F , we obtain the skew

diagram F/D

(c) If we put a positive integer in each box of the skew diagram F/D, then we

obtain a skew tableau T and say that the shape of T is F/D

(d) If the entries of the skew tableau T is taken from {1, 2, , m} and αj of them

are j for 1 ≤ j ≤ m, then we say T has content α = (α1, , αm)

(e) A skew tableau T is called semistandard if the entries in each row of T

weakly increase from left to right, and the entries in each column of T strictly

increase from top to bottom

Notation: We shall denote the set of all semistandard tableaux of shape F/D and

content α by SST(F/D, α)

Lemma 2.3.3 ([Fu]) There is a bijection between SST(F/D, α) and the set of

sequences D0 v D1 v D2 v · · · v Dh of Young diagrams such that D0 = D,

Dh = F and |Ds−1| + αs= |Ds| for all 1 ≤ s ≤ h

In fact, the semistandard tableau T which corresponds to the sequence D0 v D1 v

D2 v · · · v Dh is defined as follows: We regard F/D as a union of Ds/Ds−1 for

1 ≤ s ≤ h, and put the number s in all the boxes in Ds/Ds−1 Then T is theresulting skew tableau For example, the sequence of Young diagrams

corresponds to the semistandard tableau

1 1 22

2

1 2 21

2

1 2 22

1

2 2 21

2.3 Generalized Iterated Pieri Rules for GLn 15

where Ds = (d(s)1 , d(s)2 , , d(s)n ) for 1 ≤ s ≤ h Each entry d(s)t has (at most) 4 entries

which are adjacent to it:

d(s−1)t−1 d(s−1)t

d(s)t

d(s+1)t d(s+1)t+1The interlacing condition implies that

d(s−1)t−1 ≥ d(s)t ≥ d(s−1)t and d(s+1)t ≥ d(s)t ≥ d(s+1)t+1 That is, the two entries to the left of d(s)t are larger than or equal to d(s)t and the two

entries to the right of d(s)t are less than or equal to d(s)t These conditions are similar

to those which define a Gelfand-Tsetlin (GT) pattern In view of this, we also

call the pattern (2.7) a Gelfand-Tsetlin (GT) pattern Later, we will consider other

variants of these patterns and allow the entries to take integer values We will again

call them GT patterns

Lemma 2.3.4 There is bijection between SST(F/D, α) and the set of all the GT

patterns of the form (2.7) with nonnegative integer entries satisfying

1) D = (d(0)1 , d(0)2 , , d(0)n ), F = (d(h)1 , d(h)2 , , d(h)n ) and

2) αs =Pn

t=1d(s)t −Pn

t=1d(s−1)t for 1 ≤ s ≤ h

Proof This follows from Lemma 2.3.3

Recall that for Young diagram D = (d1, , dn), |D| = d1+ d2+ + dnis the total

number of boxes in D We now extend this notation to the elements of Λ+

n: For

λ = (λ1, , λn) ∈ Λ+

n, let

The following is a more general version of the Pieri rule, which can be considered as

folklore Since we could not find an explicit reference, we give a proof here

Theorem 2.3.5 (Generalized Pieri Rule) Let λ ∈ Λ+n and α ∈ Z≥0 Then

D v F ,

f1 ≥ d1 ≥ f2 ≥ d2 ≥ ≥ fn ≥ dn,

so that

f1+ λn≥ d1+ λn ≥ f2 + λn ≥ d2+ λn≥ ≥ fn+ λn≥ dn+ λn,that is,

µ1 ≥ λ1 ≥ µ2 ≥ λ2 ≥ ≥ µn≥ λn.This shows that λ v µ Moreover, Since

2.3 Generalized Iterated Pieri Rules for GLn 17

|λ| + α = |µ| This proves (a)

(b) By Part (a),

ρλn⊗ ρ(α)∗

n ∼= ρλ∗

n ⊗ ρ(α) n

We now let µ := µ0∗ Then

By iterating the formulas in Theorem 2.3.5, we obtain the following theorem

Theorem 2.3.6 (Generalized Iterated Pieri Rule) Let λ ∈ Λ+

n, α = (α1, , αh) ∈

Zh≥0 and β = (β1, , βl) ∈ Zl≥0

=MKµ/λ,αρµn,where the multiplicity Kµ/λ,α is equals to the number of sequences

λ(0) v λ(1) v λ(2) v v λ(h)

satisfying

1) λ = λ(0), µ = λ(h) and2) |λ(s−1)| + αs= |λ(s)| for 1 ≤ s ≤ h

Equivalently, Kµ/λ,α is the number of the GT patterns of the form

µ(0) w µ(1) w µ(2) w w µ(l)

satisfying

1) λ = µ(0), µ = µ(l) and

2.3 Generalized Iterated Pieri Rules for GLn 19

Equivalently, Kµ/λ,(α,−β) is the number of the GT patterns of the form

t=1λ(s−1)t −Pn

t=1λ(s)t for 1 ≤ s ≤ h, andii) −βj =Pn

2.4 Posets and Hibi Cones 21

In this subsection, we review some basic definitions related to posets A standard

reference is [Stan1]

Definition 2.4.1 Let Γ be a nonempty set A binary relation  on Γ is called a

partial ordering if the following conditions are satisfied:

1) For all x ∈ Γ, x  x (reflexivity)

2) If x  y and y  x, then x = y (antisymmetry)

3) If x  y and y  z, then x  z (transitivity)

A partially ordered set (poset ) is a nonempty set with a partial ordering

Notation: If x  y and x 6= y, then we write x ≺ y

Definition 2.4.2

(a) A subposet of Γ is a nonempty subset S of Γ together with the partial ordering

inherited from Γ That is, for x, y ∈ S, x  y in S if and only if x  y in Γ

(b) A chain of a poset Γ is a subposet C of Γ in which any two elements are

Definition 2.4.3.

(a) The dual of a poset Γ is the poset Γ∗ with the same underlying set Γ suchthat x  y in Γ∗ if and only if y  x in Γ

(b) Let P and Q be two posets

(i) The direct sum of P and Q is the poset P + Q on the union P ∪ Q suchthat x  y in P + Q if and only if

1) x, y ∈ P and x  y in P , or2) x, y ∈ Q and x  y in Q

(ii) The ordinal sum of P and Q is the poset P ⊕ Q on P ∪ Q such that

x  y in P ⊕ Q if and only if1) x, y ∈ P and x  y in P ,2) x, y ∈ Q and x  y in Q, or3) x ∈ P and y ∈ Q

(iii) The direct product of P and Q is the poset P × Q on the set

{(x, y) : x ∈ P and y ∈ Q}

such that (x, y)  (x0, y0) in P × Q if and only if x  x0 in P and y  y0

in Q

In this subsection, we define an increasing subset and a decreasing subset of a poset.Definition 2.4.4 ([Ho05]) Let Γ be a poset

(a) A subset S of Γ is called increasing if for any x ∈ S and any y ∈ Γ,

y  x ⇒ y ∈ S

The collection of all increasing subsets of Γ is denoted by J∗(Γ, )

2.4 Posets and Hibi Cones 23

(b) A subset S0 of Γ is called decreasing if for any x ∈ S0 and any y ∈ Γ,

y  x ⇒ y ∈ S0

The collection of all decreasing subsets of Γ is denoted by J (Γ, )

Lemma 2.4.1 Let Γ be a poset The following statements are equivalent

(a) S ∈ J∗(Γ, )

(b) Γ \ S ∈ J (Γ, )

(c) Γ \ S ∈ J∗(Γ∗, )

Proof This is clear

Note that both J∗(Γ, ) and J (Γ, ) are posets, where the partial ordering is defined

by set inclusion, i.e., S1  S2 if and only if S1 ⊆ S2 It is also clear that J∗(Γ, )

and J (Γ, ) are close under union and intersection In fact, each of them forms a

distributive lattice [Stan1]

In this subsection, we introduce the notion of a Hibi cone and review its structure

Definition 2.4.5 Let Γ be a poset and B be a nonempty subset of R

(a) A map f : Γ → B is called order preserving if f (x) ≥ f (y) for x  y

(b) A map g : Γ → B is called order reversing if g(x) ≤ g(y) for x  y

We shall denote the set of all order preserving maps from Γ to B by BΓ,

Lemma 2.4.2 If B is a subsemigroup of R containing 0, that is, it is closed underaddition, then BΓ, form a semigroup under addition of functions.

Proof First note that the zero function is in BΓ, Let f , g ∈ BΓ, If x and y in

Γ such that x  y, then

Clearly, ZΓ,≥0 is contained in ZΓ, The semigroup ZΓ,≥0 is called a Hibi cone This

is because the semigroup algebra generated by ZΓ,≥0 is called a Hibi algebra [Ho05][Hi] Moreover, ZΓ,≥0 can be identified with the set of integral points in a rationalconvex polyhedral cone [HKL]

Proposition 2.4.3 The product of two Hibi cones is a Hibi cone

Proof Let Γ1 and Γ2 be two posets We want to prove that

ZΓ≥01,× ZΓ2 ,

≥0 ∼= ZΓ1 +Γ 2 ,

as a semigroup

2.4 Posets and Hibi Cones 25

We claim that φf1,f2 is an element of ZΓ1 +Γ 2 ,

Γ1 + Γ2 and x  y By the definition of the direct sum of two posets, either x,

y ∈ Γ1 or x, y ∈ Γ2 Without lost of generality, assume that x and y ∈ Γ1 Then

φf1,f2(y) = f1(y) ≥ f1(x) = φf1,f2(x)

This proves the claim

We now define a map φ : ZΓ1 ,

Definition 2.4.6 For any subset S of Γ, the indicator function of S is the map

The following proposition gives us the relation between increasing subsets of Γ and

the order preserving maps in ZΓ,≥0

Proposition 2.4.4

(a) For f ∈ ZΓ,≥0 , the support of f

is an increasing subset of Γ

(b) If S is an increasing subset of Γ, then χS is in ZΓ,≥0

Proof (a) Let x and y be two elements of Γ Assume x ∈ Supp(f ) and x ≺ y.Then f (x) > 0 Since f ∈ ZΓ,≥0 , f (y) ≥ f (x) Therefore y ∈ Supp(f )

(b) This is clear

The Hibi cone ZΓ,≥0 has a very simple semigroup structure It turns out that theindicator functions of the increasing subsets of Γ generate ZΓ,≥0 Moreover, for twoincreasing sets S1 and S2 of Γ, we have the relation

Note that if S1 ⊆ S2, then the relation is trivial

Theorem 2.4.5 ([Ho05]) The semigroup ZΓ,≥0 is generated by {χS : S ∈ J∗(Γ, )}.More precisely, every nonzero element f of ZΓ,≥0 has a unique expression

partic-2.4.4 The Hibi Cone ZΓn,k,h ,

≥0

We revisit GT patterns in this subsection We shall show that Type I GT-patternswith nonnegative integer entries can be identified with the elements of a Hibi cone

2.4 Posets and Hibi Cones 27

Let n, k, h be positive integers We shall define a poset (Γn,k,h, ) in which the

elements can be thought as the placeholders for the entries of a GT-pattern The

underlying set is

Γn,k,h = {ηt(s): 1 ≤ t ≤ min{n, k + s}, 0 ≤ s ≤ h} (2.15)

and the partial ordering on it is defined by the interlacing conditions

ηt−1(s−1) ηt(s) η(s−1)t (2.16)

for every s and t There are two ways to visualize this poset The first way is to

arrange the elements of the poset in a similar way as a GT-pattern:

For η(s)t and η(st00) ∈ Γn,k,h, it is easy to check that ηt(s)  η(st00) in Γn,k,h if and only ifψ(η(s)t )  ψ(ηt(s00)) in Z2 ([HKL], [HL12]).

≥0.Lemma 2.4.6 There is a bijection T → fT from SST(F/D, α) to (ZΓn,k,h ,

≥0 )F,D,α.Consequently,

KF /D,α = #(ZΓn,k,h ,

≥0 )F,D,α.Here, for a set S, #(S) is its cardinality