Monomial crystals and promotion operators The monomial crystal

Monomial crystals and promotion operators

The monomial crystalM(Y_1,1Y_1,−1Y_0,2⁻¹Y_0,0⁻¹)

Mathieu Mansuy

We describe the monomial crystalM=M(Y_1,1Y_1,−1Y_0,2⁻¹Y_0,0⁻¹) of typeA₃used inarXiv :1207.3299. Letφ:M → Mbe defined by

φ(Y

Y_i,n^ui,n) =Y

Y_i+1,n+1^ui,n . We give

the sub-I_j-crystalsM_j =M_Ij(φ^j(Y_1,1Y1,−1Y_0,2⁻¹Y_0,0⁻¹))for j =0, . . . ,4,

the corresponding Dynkin diagram,

theU_q^v,j(sl₄^tor)-module associated toM_j whoseq-character is

Ξ^j( X

m∈M_j

m).

The action of the promotion operatorφonMcan be viewed as a screwing.

Y_1,1Y_1,−1Y_0,2⁻¹Y_0,0⁻¹

1

Y_1,−1Y_3,5⁻¹Y_0,4Y_0,0⁻¹ 1RR)) RR

RR ^{Y3,5−1Y3,3−1Y0,4Y0,2}

Y_1,1⁻¹Y_2,0Y_3,5⁻¹Y_0,4 2RR((

RR RR

Y_1,3⁻¹Y_1,−1Y_2,2Y_0,0⁻¹

1

2SS)) SS

SS ^{Y2,2−1Y3,5−1Y3,1Y0,4}

3

OO

Y_1,−1Y_2,4⁻¹Y_3,3Y_0,0⁻¹ 1SS)) SS SS

3

AA

Y_1,3⁻¹Y_1,1⁻¹Y_2,2Y_2,0 ² //Y_1,1⁻¹Y_2,4⁻¹Y_2,0Y_3,3 ² //

3

BB

Y_2,4⁻¹Y_2,2⁻¹Y_3,3Y_3,1 3

OO

0

1 ^//2 ^//3

AssociatedU_q^v,0(sl₄^tor)-module : V(Ξ⁰(Y_1,1Y_1,−1Y_0,2⁻¹Y_0,0⁻¹))

Y_1,5Y_1,3Y_0,4⁻¹Y_0,6⁻¹ oo ⁰ Y_1,5Y_3,3⁻¹Y_0,2Y_0,6⁻¹ oo ⁰ Y_3,5⁻¹Y_3,3⁻¹Y_0,4Y_0,2

Y_1,1⁻¹Y_2,0Y_3,5⁻¹Y_0,4 2RR((

RR RR

qqcccccccccccccccccccc0 cccccc

Y_1,1⁻¹Y_1,5Y_2,0Y_0,6⁻¹ 2RR((

RR

RR ^{Y2,2−1Y3,5−1Y3,1Y0,4}

3

OO

qqcccccccccccccccccccc0 cccccc

Y_1,5Y_2,2⁻¹Y_3,1Y_0,6⁻¹ 3

BB

Y_1,3⁻¹Y_1,1⁻¹Y_2,2Y_2,0 ² //Y_1,1⁻¹Y_2,4⁻¹Y_2,0Y_3,3 ² //

3

BB

Y_2,4⁻¹Y_2,2⁻¹Y_3,3Y_3,1 3

OO

0

1 2 ^//3

^^===

==== AssociatedU_q^v,1(sl₄^tor)-module : V(Ξ¹(Y_1,3⁻¹Y_1,1⁻¹Y_2,2Y_2,0))

Y_1,5Y_1,3Y_0,4⁻¹Y_0,6⁻¹

1

Y_1,5Y_3,3⁻¹Y_0,2Y_0,6⁻¹

oo 0

1RR((

RR

RR ^{Y3,5−1Y3,3−1Y0,4Y0,2} oo 0

Y_1,7⁻¹Y_2,6Y_3,3⁻¹Y_0,2

qqcccccccccccccccccccc0 cccccc

Y_1,7⁻¹Y_1,3Y_2,6Y_0,4⁻¹

1

Y_2,2⁻¹Y_3,5⁻¹Y_3,1Y_0,4 3

OO

qqcccccccccccccccccccc0 cccccc

Y_1,5Y_2,2⁻¹Y_3,1Y_0,6⁻¹ 3

BB

1RR((

RR RR

Y_1,7⁻¹Y_1,5⁻¹Y_2,6Y_2,4 Y_1,7⁻¹Y_2,2⁻¹Y_2,6Y_3,1 3

BB

Y_2,4⁻¹Y_2,2⁻¹Y_3,3Y_3,1 3

OO

0

1 2 3

^^===

==== AssociatedU_q^v,2(sl₄^tor)-module : V(Ξ²(Y_2,4⁻¹Y_2,2⁻¹Y_3,3Y_3,1))

Y_1,5Y_1,3Y_0,4⁻¹Y_0,6⁻¹

1

Y_1,5Y_3,3⁻¹Y_0,2Y_0,6⁻¹

oo 0

1RR((

RR

RR ^{Y3,5−1Y3,3−1Y0,4Y0,2} oo 0

Y_1,7⁻¹Y_2,6Y_3,3⁻¹Y_0,2

qqcccccccccccccccccccc0 cccccc 2

((R

RR RR R

Y_1,7⁻¹Y_1,3Y_2,6Y_0,4⁻¹

1

2RR((

RR

RR ^{Y2,8−1Y3,3−1Y3,7Y0,2}

qqcccccccccccccccccccc0 cccccc

Y_1,3Y_2,8⁻¹Y_3,7Y_0,4⁻¹ 1RR((

RR RR

Y_1,7⁻¹Y_1,5⁻¹Y_2,6Y_2,4 ² //Y_1,5⁻¹Y_2,8⁻¹Y_2,4Y_3,7 ² //Y_2,8⁻¹Y_2,6⁻¹Y_3,7Y_3,5

0

1 ^//2 3

AssociatedU_q^v,3(sl₄^tor)-module : V(Ξ³(Y_3,5⁻¹Y_3,3⁻¹Y_0,4Y_0,2))

Y_1,5Y_1,3Y_0,4⁻¹Y_0,6⁻¹

1

Y_1,3Y_3,9⁻¹Y_0,8Y_0,4⁻¹ 1RR((

RR

RR ^{Y3,9−1Y3,7−1Y0,8Y0,6}

Y_1,5⁻¹Y_2,4Y_3,9⁻¹Y_0,8 2RR((

RR RR

Y_1,7⁻¹Y_1,3Y_2,6Y_0,4⁻¹

1

2RR((

RR

RR ^{Y2,6−1Y3,9−1Y3,5Y0,8}

3

OO

Y_1,3Y_2,8⁻¹Y_3,7Y_0,4⁻¹ 1RR((

RR RR

3

BB

Y_1,7⁻¹Y_1,5⁻¹Y_2,6Y_2,4 ² //Y_1,5⁻¹Y_2,8⁻¹Y_2,4Y_3,7 ² //

3

BB

Y_2,8⁻¹Y_2,6⁻¹Y_3,7Y_3,5 3

OO

0

1 ^//2 ^//3

AssociatedU_q^v,0(sl₄^tor)-module : V(Ξ⁰(Y_1,5Y_1,3Y_0,4⁻¹Y_0,6⁻¹))

Link with thePcl-crystal B(2Λ₁)

Let us give theP_cl-crystalB(2Λ₁)of typeA⁽¹⁾₃ obtained by promotion operatorpr. In the crystal, i j= i j for alli,j.

11

1

0 14

oo

1

))S

SS SS SS SS

S ⁰ 44

oo

24 2

))S

SS SS SS SS S

qqcccccccccccccccccccccc0 cccccccccccccc

12

1

2

))S

SS SS SS SS

S 34

3

OO

qqcccccccccccccccccccccc0 cccccccccccccc

13 1

))S

SS SS SS SS S

3

@@

22 ^{2 //}23 ^{2 //}

3

@@

33

3

OO

0

1 ^//2 ^//3

^^===

==== Crystal graph of the U_q( ˆsl₄)-moduleW(2$₁)