The inequalities defining polyhedral realizations and monomial realizations of crystal bases

The inequalities defining polyhedral realizations and monomial realizations of crystal bases

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima) RIMS

共同研究「組合せ論的表現論の最近の進展」

October 8, 2020

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 1 / 51

Introduction

Two main objects of this talk:

Polyhedral realizations of crystal bases B(∞).

Monomial realizations of crystal bases B(λ) (λ : dominant integral weight).

Polyhedral realizations of crystal basesB(∞)

ι:= (· · · ,i₃,i₂,i₁) : infinite sequence of indices in I of a symmetrizable K.M alg. g s.t.

i_k ̸=i_k+1 (k ∈Z>0),

♯{k ∈Z>0|i_k =j}=∞ (for anyj ∈I).

ex)g: rank 2, ι= (· · · ,2,1,2,1,2,1).

ex)g: rank 3, ι= (· · · ,3,1,2,3,2,1,3,1,2,3,2,1).

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 2 / 51

Introduction : Polyhedral realizations of B ( ∞ )

One can define a crystal structure on

Z^∞={(· · · ,a₃,a₂,a₁)|a_l ∈Z, a_k = 0 (k ≫0)} associated with ι and denote it byZ^∞_ι .

Fact(Kashiwara, Nakashima-Zelevinsky)

For eachι, there exists an embedding of crystals Ψι: Ψ_ι :B(∞),→Z^∞ι .

Im(Ψ_ι) (∼=B(∞)) : Polyhedral realization of B(∞)

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 3 / 51

Example : Polyhedral realizations of B ( ∞ )

g: typeA₂, ι= (· · · ,2,1,2,1,2,1).

The crystal graph of (B(∞)∼=)Im(Ψ_ι)⊂Z^∞ is as follows : (...,0,0,0)

(...,0,1,0) (...,0,0,1)

(...,0,0,2) (...,0,1,1) (...,1,1,0) (...,0,2,0) (...,0,0,3) (...,0,1,2) (...,0,2,1) (...,1,1,1) (...,1,2,0)

2

((

1

vv

1

ww ^{2 1 2} ''

2 xx ¹

1

xx ^{2 1 12} &&

The setIm(Ψ_ι) coincides with the set of (· · · ,a₃,a₂,a₁)∈Z^∞ s.t.

a1 ≥0, a2 ≥a3 ≥0, ak = 0 (k >3).

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 4 / 51

Introduction : Polyhedral realizations of B ( ∞ )

Problem

Find an explicit form of inequalities defining Im(Ψ_ι).

Nakashima-Zelevinsky(1997)

Invention of polyhedral realizations and algorithm calculating inequalities definingIm(Ψ_ι) (under a condition).

In the case g : A_n or A⁽¹⁾n -type or rank2 K.M alg, and

ι= (· · · ,n,· · · ,2,1,n,· · · ,2,1), an explicit form of inequalities defining Im(Ψ_ι) is given.

Hoshino(2005), Kim-Shin(2008)

g:simple, ι= (· · · ,n,· · · ,1,n,· · · ,1) ⇒an explicit form of the inequalities ofIm(Ψ_ι)

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 5 / 51

Introduction : Polyhedral realizations of B ( ∞ )

Littlemann(1998)

We suppose that g is a fin. dim. simple. Lie alg.

i= (i_N,· · · ,i₁) : a reduced word of the long. ele. w₀ ∈W. ι= (· · · ,i_N,· · · ,i₁),

⇒Im(Ψ_ι)⊂ {(· · · ,a_N,· · · ,a₂,a₁)∈Z^∞|a_N+1 =a_N+2 =· · ·= 0}.

∴We can regard as Im(Ψ_ι)⊂Z^N.

Im(Ψ_ι)⊂Z^N coincides with the set of integer points of the

Littelmann’s String coneS(i1,···,iN)⊂R^N : Im(Ψ_ι) =S(i1,···,iN)∩Z^N. i: ‘nice decomposition’ ⇒an explicit form of string cone S(i) : type A : i= (1,2,1,3,2,1,· · · ,n,· · · ,1),

type B,C :i=

(1,(2,1,2),(3,2,1,2,3),· · · ,(n,n−1· · · ,2,1,2,· · · ,n−1,n)).

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 6 / 51

Introduction : Monomial realizations of B (λ)

Monomial realizations of crystal bases B(λ) (Nakajima, Kashiwara) Each element of crystal base B(λ) is realized as a Laurent monomial of doubly indexed variables{Xs,i|s ∈Z, i ∈I}.

ex)B(Λ₁) of typeA₃

X_1,1 →^f˜1 X_1,2 X_2,1

f˜2

→ X_1,3 X_2,2

f˜3

→ 1

X_2,3. Main results

Explicit forms of inequalities defining polyhedral realizations of B(∞) in terms of rectangular tableaux for ‘adapted’ sequence ι and classical Lie algebrag.

A conjecture that inequalities for B(∞) are obtained from monomial realizations of B(λ) via tropicalization.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 7 / 51

1. Polyhedral realizations of B( ∞ )

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 8 / 51

A morphism of crystals

Definition

Amorphism ψ :B1 → B2 of crystals B1, B2 is a map B1

⊔{0} → B2

⊔{0}s.t. ψ(0) = 0 and forb ∈ B1, (1) wt(ψ(b)) = wt(b) if ψ(b)̸= 0,

(2) ε_i(ψ(b)) =ε_i(b) if ψ(b)̸= 0, (3) φ_i(ψ(b)) = φ_i(b) ifψ(b)̸= 0,

(4) ψ(˜e_i(b)) = ˜e_iψ(b) if ψ(b)̸= 0 andψ(˜e_i(b))̸= 0, (5) ψ( ˜f_i(b)) = ˜f_iψ(b) if ψ(b)̸= 0 andψ( ˜f_i(b))̸= 0,

fori ∈I.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 9 / 51

A strict morphism of crystals

Definition

Astrict morphism ψ :B1 → B2 of crystals B1, B2 is a map B1

⊔{0} → B2

⊔{0}s.t. ψ(0) = 0 and forb ∈ B1, (1) wt(ψ(b)) = wt(b) if ψ(b)̸= 0,

(2) ε_i(ψ(b)) =ε_i(b) if ψ(b)̸= 0, (3) φi(ψ(b)) = φi(b) ifψ(b)̸= 0, (4) ψ(˜e_i(b)) = ˜e_iψ(b),

(5) ψ( ˜f_i(b)) = ˜f_iψ(b), fori ∈I.

An injective strict morphism is said to be strict embedding.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 10 / 51

A crystal structure of Z

Z^∞ :={x= (· · · ,x₄,x₃,x₂,x₁)|x_k ∈Z, x_l = 0(l ≫0)}. ι:= (· · · ,i₃,i₂,i₁) : infinite sequence of I

s.t. i_k ̸=i_k+1 (k ∈Z>0) and ♯{k ∈Z>0|i_k =j}=∞ (for anyj ∈I).

We define a crystal str. onZ^∞ ass. to ι as follows:

wt(x) :=− ∑

j∈Z≥1

x_jα_ij,

σk(x) :=xk +∑

j>k

⟨hi_k, αi_j⟩xj (k ∈Z≥1, x∈Z^∞), εi(x) :=maxk∈Z≥1; i_k=iσk(x), φi(x) :=⟨wt(x),hi⟩+εi(x) (i ∈I).

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 11 / 51

A crystal structure of Z

Z^∞ :={x= (· · · ,x₄,x₃,x₂,x₁)|x_k ∈Z, x_l = 0(l ≫0)}. ι:= (· · · ,i₃,i₂,i₁) : infinite sequence of I

s.t. i_k ̸=i_k+1 (k ∈Z>0) and ♯{k ∈Z>0|i_k =j}=∞ (for anyj ∈I).

We define a crystal str. onZ^∞ ass. to ι as follows:

wt(x) :=− ∑

j∈Z≥1

x_jα_ij, σk(x) :=xk +∑

j>k

⟨hi_k, αi_j⟩xj (k ∈Z≥1, x∈Z^∞), εi(x) :=maxk∈Z≥1; i_k=iσk(x), φi(x) :=⟨wt(x),hi⟩+εi(x) (i ∈I).

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 11 / 51

A crystal structure of Z

Forx= (x_k)_k∈Z≥1 ∈Z^∞ and i ∈I,

( ˜f_i(x))_k :=x_k +δ_k,mi, (˜e_i(x))_k :=x_k −δ_k,m′

i if ε_i(x)>0, (˜e_i(x)) = 0 if ε_i(x) = 0, wherem_i, m_i^′ are combinatorially determined from {σ_k(x)}k∈Z≥1. Theorem (Nakashima-Zelevinsky)

(Z^∞, ˜e_i, ˜f_i, ε_i, φ_i, wt) is a crystal. We denote it byZ^∞ι .

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 12 / 51

Polyhedral realizations

Theorem (Nakashima-Zelevinsky)

There is a unique strict embedding of crystals Ψ_ι :B(∞),→Z^∞ι s.t.

Ψ_ι(u_∞) = (· · · ,0,0,0). Here u_∞ is the highest weight vector of B(∞).

Definition

ImΨ_ι(∼=B(∞)) is called a polyhedral realizationof B(∞).

Problem

Find explicit forms inequalities definingIm(Ψ_ι).

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 13 / 51

2. Calculations of Polyhedral realizations

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 14 / 51

Polyhedral realizations of B ( ∞ )

Nakashima and Zelevinsky found a way calculating the

inequalities definingIm(Ψ_ι) if ι satisfies apositivity condition.

Defining a set Ξ_ι ⊂Hom_Z(Z^∞,Z), they described Im(Ψ_ι) as Im(Ψι) = {x∈Z^∞|φ(x)≥0, ∀φ∈Ξι}.

Let us see the positivity condition and a construction of Ξ_ι.

Forι= (· · · ,i₃,i₂,i₁) and k ∈Z≥1,

k⁻ :=max({l ∈Z_≥1|l <k, i_k =i_l} ∪ {0}), k⁺ :=min{l ∈Z≥1|l >k, i_k =i_l}.

ex)ι= (· · · ,2,1,2,1,2,1)⇒1⁻ = 0, 2⁻= 0, 3⁻ = 1, 4⁻= 2, 5⁻ = 3, 6⁻ = 4, 1⁺ = 3, 2⁺= 4, 3⁺ = 5, 4⁺= 6.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 15 / 51

Polyhedral realizations of B ( ∞ )

Nakashima and Zelevinsky found a way calculating the

inequalities definingIm(Ψ_ι) if ι satisfies apositivity condition.

Defining a set Ξ_ι ⊂Hom_Z(Z^∞,Z), they described Im(Ψ_ι) as Im(Ψι) = {x∈Z^∞|φ(x)≥0, ∀φ∈Ξι}.

Let us see the positivity condition and a construction of Ξ_ι. Forι= (· · · ,i₃,i₂,i₁) and k ∈Z≥1,

k⁻ :=max({l ∈Z_≥1|l <k, i_k =i_l} ∪ {0}), k⁺ :=min{l ∈Z≥1|l >k, i_k =i_l}.

ex)ι= (· · · ,2,1,2,1,2,1)⇒1⁻ = 0, 2⁻= 0, 3⁻ = 1, 4⁻= 2, 5⁻ = 3, 6⁻= 4, 1⁺ = 3, 2⁺= 4, 3⁺ = 5, 4⁺= 6.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 15 / 51

Calculations of Polyhedral realization of B ( ∞ )

Fork ∈Z≥1, we define x_k ∈Hom_Z(Z^∞,Z) as x_k(· · · ,a₃,a₂,a₁) := a_k.

Usingx_k, each φ∈Hom_Z(Z^∞,Z) can be written as

φ=∑

c_kx_k, c_k ∈Z.

Letβ_k ∈Hom_Z(Z^∞,Z) (k ∈Z_≥1) be β_k =x_k + ∑

k<j<k⁺

⟨h_ik, α_ij⟩x_j +x_k⁺.

Forφ=∑

c_kx_k ∈Hom_Z(Z^∞,Z) and k ∈Z_≥1, we define S_k(φ)∈Hom_Z(Z^∞,Z) as

S_k(φ) := {

φ−c_kβ_k if c_k ≥0, φ−c_kβ_k− if c_k <0.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 16 / 51

Calculations of Polyhedral realization of B ( ∞ )

Fork ∈Z≥1, we define x_k ∈Hom_Z(Z^∞,Z) as x_k(· · · ,a₃,a₂,a₁) := a_k.

Usingx_k, each φ∈Hom_Z(Z^∞,Z) can be written as

φ=∑

c_kx_k, c_k ∈Z. Letβ_k ∈Hom_Z(Z^∞,Z) (k ∈Z_≥1) be

β_k =x_k + ∑

k<j<k⁺

⟨h_ik, α_ij⟩x_j +x_k⁺.

Forφ=∑

c_kx_k ∈Hom_Z(Z^∞,Z) and k ∈Z_≥1, we define S_k(φ)∈Hom_Z(Z^∞,Z) as

S_k(φ) := {

φ−c_kβ_k if c_k ≥0, φ−c_kβ_k− if c_k <0.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 16 / 51

Calculations of Polyhedral realization of B ( ∞ )

Fork ∈Z≥1, we define x_k ∈Hom_Z(Z^∞,Z) as x_k(· · · ,a₃,a₂,a₁) := a_k.

Usingx_k, each φ∈Hom_Z(Z^∞,Z) can be written as

φ=∑

c_kx_k, c_k ∈Z. Letβ_k ∈Hom_Z(Z^∞,Z) (k ∈Z_≥1) be

β_k =x_k + ∑

k<j<k⁺

⟨h_ik, α_ij⟩x_j +x_k⁺.

Forφ=∑

c_kx_k ∈Hom_Z(Z^∞,Z) and k ∈Z_≥1, we define S_k(φ)∈Hom_Z(Z^∞,Z) as

S_k(φ) :=

{

φ−c_kβ_k if c_k ≥0, φ−c_kβ_k− if c_k <0.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 16 / 51

Polyhedral realization of B ( ∞ )

Forφ=∑

c_kx_k ∈Hom_Z(Z^∞,Z), S_k(φ) :=

{

φ−c_kβ_k if c_k ≥0, φ−c_kβ_k− if c_k <0, where we set β₀ := 0. We also define

Ξ_ι :={S_jl · · ·S_j1x_j0|l ≥0,j₀,j₁,· · · ,j_l ≥1}.

Positivity condition

For anyφ=∑

ckxk ∈Ξι, if k⁻ = 0 (k ∈Z≥1) thenck ≥0. Theorem (Nakashima-Zelevinsky)

If ι satisfies the positivity condition then

Im(Ψ_ι)(∼=B(∞)) ={x∈Z^∞|φ(x)≥0, ∀φ∈Ξ_ι}.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 17 / 51

Polyhedral realization of B ( ∞ )

Forφ=∑

c_kx_k ∈Hom_Z(Z^∞,Z), S_k(φ) :=

{

φ−c_kβ_k if c_k ≥0, φ−c_kβ_k− if c_k <0, where we set β₀ := 0. We also define

Ξ_ι :={S_jl · · ·S_j1x_j0|l ≥0,j₀,j₁,· · · ,j_l ≥1}.

Positivity condition

For anyφ=∑

ckxk ∈Ξι, if k⁻ = 0 (k ∈Z≥1) thenck ≥0.

Theorem (Nakashima-Zelevinsky)

If ι satisfies the positivity condition then

Im(Ψ_ι)(∼=B(∞)) ={x∈Z^∞|φ(x)≥0, ∀φ∈Ξ_ι}.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 17 / 51

An example of the polyhedral realization of B ( ∞ )

Example)g : type A₂, ι= (· · · ,2,1,2,1,2,1).

1⁻= 2⁻= 0, k⁻ >0 (k >2).

We rewrite a vector (· · · ,x₆,x₅,x₄,x₃,x₂,x₁) as

(· · ·,x_3,2,x_3,1,x_2,2,x_2,1,x_1,2,x_1,1). (x_l,1 =x_2l−1, x_l,2 =x_2l) Similarly, we rewrite S_l,1 =S_2l−1, S_l,2 =S_2l.

Recall) positivity condition⇔ the coefficients of x₁ =x_1,1 and x₂ =x_1,2 in each φ∈Ξ_ι are non-negative.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 18 / 51

An example of the polyhedral realization of B ( ∞ )

Example)g : type A₂, ι= (· · · ,2,1,2,1,2,1).

1⁻= 2⁻= 0, k⁻ >0 (k >2).

We rewrite a vector (· · · ,x₆,x₅,x₄,x₃,x₂,x₁) as

(· · ·,x_3,2,x_3,1,x_2,2,x_2,1,x_1,2,x_1,1). (x_l,1 =x_2l−1, x_l,2 =x_2l) Similarly, we rewrite S_l,1 =S_2l−1, S_l,2 =S_2l.

Recall) positivity condition⇔ the coefficients of x₁ =x_1,1 and x₂ =x_1,2 in each φ∈Ξ_ι are non-negative.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 18 / 51

An example of the polyhedral realization of B ( ∞ )

Example)g : type A₂, ι= (· · · ,2,1,2,1,2,1).

1⁻= 2⁻= 0, k⁻ >0 (k >2).

We rewrite a vector (· · · ,x₆,x₅,x₄,x₃,x₂,x₁) as

(· · ·,x_3,2,x_3,1,x_2,2,x_2,1,x_1,2,x_1,1). (x_l,1 =x_2l−1, x_l,2 =x_2l) Similarly, we rewrite S_l,1 =S_2l−1, S_l,2 =S_2l.

Recall) positivity condition⇔ the coefficients of x₁ =x_1,1 and x₂ =x_1,2 in each φ∈Ξ_ι are non-negative.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 18 / 51

An example of the polyhedral realization of B ( ∞ )

Example)g : type A₂, ι= (· · · ,2,1,2,1,2,1).

xk,1 Sk,1

⇄ xk,2 −xk+1,1 Sk,2

⇄ −xk+1,2,

Sk+1,1 Sk+1,2

x_k,2

Sk,2

⇄ x_k+1,1 −x_k+1,2

Sk+1,1

⇄ −x_k+2,1,

Sk+1,2 Sk+2,1

fork ∈Z≥1 and other actions are trivial.

Thus,

Ξ_ι ={x_k,1, x_k,2−x_k+1,1, −x_k+1,2, x_k,2, x_k+1,1−x_k+1,2,−x_k+2,1|k ≥1}. The coefficients ofx_1,1 and x_1,2 in each φ∈Ξ_ι are non-negative.

∴ ι satisfies the positivity conditionand Im(Ψ_ι) ={x∈Z^∞|φ(x)≥0, ∀φ∈Ξ_ι}

= {x∈Z^∞|x_k+1,2 =x_k+2,1 = 0 (k ≥1), x_1,2 ≥x_2,1 ≥0, x_1,1 ≥0}.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 19 / 51

An example of the polyhedral realization of B ( ∞ )

Example)g : type A₂, ι= (· · · ,2,1,2,1,2,1).

xk,1 Sk,1

⇄ xk,2 −xk+1,1 Sk,2

⇄ −xk+1,2,

Sk+1,1 Sk+1,2

x_k,2

Sk,2

⇄ x_k+1,1 −x_k+1,2

Sk+1,1

⇄ −x_k+2,1,

Sk+1,2 Sk+2,1

fork ∈Z≥1 and other actions are trivial. Thus,

Ξ_ι ={x_k,1, x_k,2−x_k+1,1, −x_k+1,2, x_k,2, x_k+1,1−x_k+1,2,−x_k+2,1|k ≥1}.

The coefficients ofx_1,1 and x_1,2 in each φ∈Ξ_ι are non-negative.

∴ ι satisfies the positivity conditionand Im(Ψ_ι) ={x∈Z^∞|φ(x)≥0, ∀φ∈Ξ_ι}

= {x∈Z^∞|x_k+1,2 =x_k+2,1 = 0 (k ≥1), x_1,2 ≥x_2,1 ≥0, x_1,1 ≥0}.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 19 / 51

An example of the polyhedral realization of B ( ∞ )

Example)g : type A₂, ι= (· · · ,2,1,2,1,2,1).

xk,1 Sk,1

⇄ xk,2 −xk+1,1 Sk,2

⇄ −xk+1,2,

Sk+1,1 Sk+1,2

x_k,2

Sk,2

⇄ x_k+1,1 −x_k+1,2

Sk+1,1

⇄ −x_k+2,1,

Sk+1,2 Sk+2,1

fork ∈Z≥1 and other actions are trivial. Thus,

Ξ_ι ={x_k,1, x_k,2−x_k+1,1, −x_k+1,2, x_k,2, x_k+1,1−x_k+1,2,−x_k+2,1|k ≥1}. The coefficients ofx_1,1 and x_1,2 in each φ∈Ξ_ι are non-negative.

∴ ι satisfies the positivity conditionand Im(Ψ_ι) ={x∈Z^∞|φ(x)≥0, ∀φ∈Ξ_ι}

= {x∈Z^∞|x_k+1,2 =x_k+2,1 = 0 (k ≥1), x_1,2 ≥x_2,1 ≥0, x_1,1 ≥0}.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 19 / 51

An example of the polyhedral realization of B ( ∞ )

Example)g : type A₂, ι= (· · · ,2,1,2,1,2,1).

xk,1 Sk,1

⇄ xk,2 −xk+1,1 Sk,2

⇄ −xk+1,2,

Sk+1,1 Sk+1,2

x_k,2

Sk,2

⇄ x_k+1,1 −x_k+1,2

Sk+1,1

⇄ −x_k+2,1,

Sk+1,2 Sk+2,1

fork ∈Z≥1 and other actions are trivial. Thus,

Ξ_ι ={x_k,1, x_k,2−x_k+1,1, −x_k+1,2, x_k,2, x_k+1,1−x_k+1,2,−x_k+2,1|k ≥1}. The coefficients ofx_1,1 and x_1,2 in each φ∈Ξ_ι are non-negative.

∴ ι satisfies the positivity conditionand Im(Ψ_ι) ={x∈Z^∞|φ(x)≥0, ∀φ∈Ξ_ι}

= {x∈Z^∞|x_k+1,2 =x_k+2,1 = 0 (k ≥1), x_1,2 ≥x_2,1 ≥0, x_1,1 ≥0}.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 19 / 51

An example which does not satisfy the positivity condition

Example)g : type A₃, ι= (· · · ,2,1,2,3,2,1).

x1 S1

→ −x5+x4+x2 S2

→ −x5+x3 S5

→ −x4 +x3−x2+x1.

Thus,−x₄+x₃−x₂+x₁ ∈Ξ_ι and 2⁻ = 0.

∴ι does not satisfy the positivity condition.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 20 / 51

An example which does not satisfy the positivity condition

Example)g : type A₃, ι= (· · · ,2,1,2,3,2,1).

x1 S1

→ −x5+x4+x2 S2

→ −x5+x3 S5

→ −x4 +x3−x2+x1. Thus,−x₄+x₃−x₂+x₁ ∈Ξ_ι and 2⁻ = 0.

∴ι does not satisfy the positivity condition.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 20 / 51

3. An explicit form of the polyhedral realization for B ( ∞ )

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 21 / 51

Infinite sequences adapted to A

A= (ai,j)i,j∈I : The Cartan matrix of g Definition

If ι satisfies the following condition, we sayι isadapted to A: Fori,j ∈I with i ̸=j and ai,j ̸= 0, the subsequence ofι consisting of i, j is

(· · · ,i,j,i,j,i,j,i,j) or (· · · ,j,i,j,i,j,i,j,i).

Example)g : type A3, ι= (· · · ,2,1,3,2,1,3,2,1,3) subsequence consisting of 1, 2 : (· · · ,2,1,2,1,2,1) subsequence consisting of 2, 3 : (· · · ,2,3,2,3,2,3) Sincea_1,3 = 0 we do not need consider the pair 1, 3. Thus,ι is adapted to A.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 22 / 51

Infinite sequences adapted to A

A= (ai,j)i,j∈I : The Cartan matrix of g Definition

If ι satisfies the following condition, we sayι isadapted to A: Fori,j ∈I with i ̸=j and ai,j ̸= 0, the subsequence ofι consisting of i, j is

(· · · ,i,j,i,j,i,j,i,j) or (· · · ,j,i,j,i,j,i,j,i).

Example)g : type A3, ι= (· · · ,2,1,3,2,1,3,2,1,3) subsequence consisting of 1, 2 : (· · · ,2,1,2,1,2,1) subsequence consisting of 2, 3 : (· · · ,2,3,2,3,2,3) Sincea_1,3 = 0 we do not need consider the pair 1, 3.

Thus,ι is adapted to A.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 22 / 51

Infinite sequences adapted to A

Example)g : type A₃, ι= (· · · ,3,2,1,3,2,3,1,2,3,1,2,1) subsequence consisting of 1, 2 : (· · · ,2,1,2,1,2,1) subsequence consisting of 2, 3 : (· · · ,3,2,3,2,3,2) ι is adapted toA.

Example)g : type A₃, ι= (· · · ,2,1,2,3,2,1)

subsequence consisting of 1, 2 : (· · · ,2,1,2,2,1) ι isnot adapted to A.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 23 / 51

Tableaux description of Ξ

In this section, let g be classical type (A_n, B_n, C_n or D_n).

In what follows, we suppose thatι is adapted to the Cartan matrix of gand consider the following problem:

Problem

Find an explicit form of inequalities defining Im(Ψ_ι).

Let us constructan explicit form of Ξ_ι by using rectangular tableaux.

Recall If ι satisfies the positivity condition then Im(Ψ_ι) = {x∈Z^∞|φ(x)≥0, ∀φ∈Ξ_ι}(∼=B(∞)).

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 24 / 51

Tableaux description of Ξ

Recall

ι isadapted to A= (aij)^def⇔

Fori,j ∈I (i ̸=j, a_i,j ̸= 0), the subsequence ofι consisting of i,j is (· · · ,i,j,i,j,i,j,i,j) or (· · · ,j,i,j,i,j,i,j,i).

Let (p_i,j)_{i̸=j, ai,j̸=0} be the set of integers s.t.

p_i,j = {

1 if (· · · ,j,i,j,i,j,i), 0 if (· · · ,i,j,i,j,i,j).

Fork (2≤k ≤n), we set P(k) :=

{

p_2,1+p_3,2+· · ·+p_n−2,n−3+p_n,n−2 if k =n, g:type D_n, p_2,1+p_3,2+p_4,3+· · ·+p_k,k−1 if o.w.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 25 / 51

Tableaux description of Ξ

Recall

ι isadapted to A= (aij)^def⇔

Fori,j ∈I (i ̸=j, a_i,j ̸= 0), the subsequence ofι consisting of i,j is (· · · ,i,j,i,j,i,j,i,j) or (· · · ,j,i,j,i,j,i,j,i).

Let (p_i,j)_{i̸=j, ai,j̸=0} be the set of integers s.t.

p_i,j = {

1 if (· · · ,j,i,j,i,j,i), 0 if (· · · ,i,j,i,j,i,j).

Fork (2≤k ≤n), we set P(k) :=

{

p_2,1+p_3,2+· · ·+p_n−2,n−3+p_n,n−2 if k =n, g:type D_n, p_2,1+p_3,2+p_4,3+· · ·+p_k,k−1 if o.w.

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 25 / 51

Tableaux descriptions

ι= (· · · ,i_k,· · · ,i₃,i₂,i₁).

Fork ∈Z≥1, we write

x_k =x_s,j, S_k =S_s,j,

if ik =j and j is appearing s times in ik, ik−1, · · · ,i1.

Example)ι= (· · · ,2,1,3,2,1,3,2,1,3)

(· · · ,x7,x6,x5,x4,x3,x2,x1) = (· · ·,x3,3,x2,2,x2,1,x2,3,x1,2,x1,1,x1,3)

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 26 / 51

Tableaux descriptions

ι= (· · · ,i_k,· · · ,i₃,i₂,i₁).

Fork ∈Z≥1, we write

x_k =x_s,j, S_k =S_s,j,

if ik =j and j is appearing s times in ik, ik−1, · · · ,i1. Example)ι= (· · · ,2,1,3,2,1,3,2,1,3)

(· · · ,x7,x6,x5,x4,x3,x2,x1) = (· · ·,x3,3,x2,2,x2,1,x2,3,x1,2,x1,1,x1,3)

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 26 / 51

Tableaux descriptions

g=A_n case

For 1≤j ≤n+ 1 and s ∈Z, we set j ^A

s

:= x

− x

∈ Hom

( Z

, Z )

j s = j ^A

s are obtained from 1

s =x_s,1 by operators S_m,j (1≤s):

1 s 2

s 3

s · · · n

s n+1

s Ss,1

++ ^Ss+P(2),2++ ^Ss+P(3),3** ++ ^Ss+P(n),n ,,ll

S_s+P(n)+1,n

hhkk

S_s+P(3)+1,3

kk

S_s+P(2)+1,2

kk

Ss+1,1

Yuki Kanakubo, University of Tsukuba (Partially joint work with Toshiki Nakashima)The inequalities defining polyhedral realizations and monomial realizations of crystal basesRIMS共同研究「組合せ論的表現論の最近の進展」October 8, 2020 27 / 51