Monomial crystals and promotion operators Description of the

Monomial crystals and promotion operators

Description of theU_q( ˆsl₄)-crystalM(Y_1,1Y_1,−1Y_0,2⁻¹Y_0,0⁻¹)

Mathieu Mansuy

We describe the crystalM_NC =M(Y_1,1Y1,−1Y_0,2⁻¹Y_0,0⁻¹)of type A₃used in Section 2.4 of the thesis. Letφ:M_NC → M_NC be the map 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_1,1Y_1,−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∈Mj

m).

The action of the promotion operatorφonM_NC can 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 1 ))

Y_3,5⁻¹Y_3,3⁻¹Y_0,4Y_0,2

Y_1,1⁻¹Y_2,0Y_3,5⁻¹Y_0,4 2 ((

Y_1,3⁻¹Y_1,−1Y_2,2Y_0,0⁻¹

1

2 ))

Y_2,2⁻¹Y_3,5⁻¹Y_3,1Y_0,4 3

OO

Y_1,−1Y_2,4⁻¹Y_3,3Y_0,0⁻¹ 1 ))

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 2 ((

qq 0 Y_1,1⁻¹Y_1,5Y_2,0Y_0,6⁻¹

2 ((

Y_2,2⁻¹Y_3,5⁻¹Y_3,1Y_0,4 3

OO

qq 0 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

1 ((

Y_3,5⁻¹Y_3,3⁻¹Y_0,4Y_0,2

oo 0

Y_1,7⁻¹Y_2,6Y_3,3⁻¹Y_0,2

qq 0 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

qq 0 Y_1,5Y_2,2⁻¹Y_3,1Y_0,6⁻¹

3

BB

1 ((

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

1 ((

Y_3,5⁻¹Y_3,3⁻¹Y_0,4Y_0,2

oo 0

Y_1,7⁻¹Y_2,6Y_3,3⁻¹Y_0,2

qq 0

2 ((

Y_1,7⁻¹Y_1,3Y_2,6Y_0,4⁻¹

1

2 ((

Y_2,8⁻¹Y_3,3⁻¹Y_3,7Y_0,2

qq 0 Y_1,3Y_2,8⁻¹Y_3,7Y_0,4⁻¹

1 ((

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 1 ((

Y_3,9⁻¹Y_3,7⁻¹Y_0,8Y_0,6

Y_1,5⁻¹Y_2,4Y_3,9⁻¹Y_0,8 2 ((

Y_1,7⁻¹Y_1,3Y_2,6Y_0,4⁻¹

1

2 ((

Y_2,6⁻¹Y_3,9⁻¹Y_3,5Y_0,8 3

OO

Y_1,3Y_2,8⁻¹Y_3,7Y_0,4⁻¹ 1 ((

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 B₀(2Λ₁)_aff

Let us give theP_cl-crystalB₀(2Λ₁)_affof typeA⁽¹⁾₃ obtained by promotion operatorpr. In the crystal, i j= i j for alli,j.

11

1

0 14

oo

1

))

0 44

oo

24 2

))

qq 0

12

1

2

))

34

3

OO

qq 0

13 1

))

3

@@

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

3

@@

33

3

OO

0

1 2 3

Crystal graph of the U_q( ˆsl₄)⁰-moduleW(2$₁)