Cluster algebras and snake modules
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1 (Lanzhou University and University of Connecticut) joint with Jian-Rong Li & Yan-Feng Luo A conference celebrating the 60th birthday of Prof Vyjayanthi Chari The Catholic University of America, Washington, DC, June 7, 2018
2 Cluster algebras Cluster algebras Representations of quantum affine algebras Snake modules Cluster algebras were introduced by Fomin and Zelevinsky around 2000
3 Cluster algebras Representations of quantum affine algebras Snake modules Example Q : 1 2 ( ) ( ( ( 1+x2 x1 ( 1+x2 x1 ( ) (x 1, x 2 ), 1 2 ) (, x 2 ), 1 2 µ 2 µ 1 ) (, x1+1+x2 ), 1 2 x1x2 µ 1 ( ( x1+1 x2 (x 1, x1+1 ( x1+1+x2 x1x2 ), x1+1+x2 ), 1 2 x1x2 µ 2 x2 ) ), 1 2 µ 1 ), x1+1 )), 1 2 x2 A(Q) = Z[x 1, x 2, 1+x2 x1, x1+1 x2, x1+x2+1 ] Q(x x1x2 1, x 2 )
4 Cluster algebras Representations of quantum affine algebras Snake modules Representations of quantum affine algebras g simple Lie algebra over C I = {1, 2,, n}, where n is the rank of g U q (ĝ) (untwisted) quantum affine algebra with quantum parameter q C not a root of unity Chari and Pressley classified the finite dimensional simple U q (ĝ)-modules in terms of Drinfeld polynomials, ie, I-tuples of polynomials (P 1,, P n ), where P i, i = 1,, n, are polynomials in one indeterminate u with coefficients in C and constant term 1
5 Cluster algebras Representations of quantum affine algebras Snake modules For example, in A n, (1,, 1, 1 au, 1, 1) is a Drinfeld polynomial of a fundamental module i (1,, 1, (1 au)(1 aq 2 u), 1, 1) is a Drinfeld i polynomial of a Kirillov-Reshetikhin module (1,, 1, (1 au)(1 aq 2 u), (1 aq 5 u)(1 aq 7 u), 1,, 1) i i+1 is a Drinfeld polynomial of a minimal affinization Minimal affinizations were introduced by Chari in 1995
6 Cluster algebras Representations of quantum affine algebras Snake modules Let P be the free abelian multiplicative group of monomials in infinitely many formal variables (Y i,a ) i I,a C and P + P denote the set of all dominant monomials Then P + {f d simple U q (ĝ)-modules} m L(m)
7 Cluster algebras Representations of quantum affine algebras Snake modules Example, in A n, Drinfeld polynomials Dominant monomials simple Uq(ĝ)-modules (1,, 1, 1 au, 1, 1) Yi,a L(Yi,a) i (1,, 1, (1 au)(1 aq 2 u), 1, 1) Yi,aY i,aq 2 L(Yi,aY i,aq 2) i (1,, 1, (1 au)(1 aq 2 u), (1 aq 5 u)(1 aq 7 u), 1,, 1) Yi,aY i,aq 2Y i+1,aq 5Y i+1,aq 7 L(Yi,aY i,aq 2Y i+1,aq 5Y i+1,aq 7) i i+1 Y 1,aq 13Y 2,aq 10Y 3,aq 7Y 2,aq 4Y 1,aq 1 L(Y 1,aq 13Y 2,aq 10Y 3,aq 7Y 2,aq 4Y 1,aq 1) Snake modules were introduced by Mukhin and Young in 2012 Snake modules corresponds to ladder representations in the setting of representations of p-adic groups Ladder representations were introduced by Lapid and Minguez in 2014 Kirillov Reshetikhin modules minimal affinizations prime snake modules snake modules
8 Cluster algebras Representations of quantum affine algebras Snake modules Frenkel and Reshetikhin introduced the theory of q-characters The q-character is an injective ring homomorphism from Grothendick ring of the category of f d simple U q (ĝ)-modules to Laurent polynomial ring Z[Y ± i,a ] i I,a C For example, g = sl 2, χ q (L(Y 1,a )) = Y 1,a + Y 1 1,aq 2 Frenkel-Mukhin algorithm can be used to compute the q-characters of a large family of modules For example, the modules whose q-characters have only one dominant monomial
9 Cluster algebras Representations of quantum affine algebras Snake modules The simple module L(Y it,aq k T Y i2,aq k 2 Y i1,aq k 1 ) is called a snake module/prime snake module/minimal snake module if for all 2 t T, the point (i t, k t ) is in snake position/prime snake position/minimal snake position with respect to (i t 1, k t 1 )
10 Cluster algebras Representations of quantum affine algebras Snake modules A point (i, k ) is said to be in snake position (respectively, prime snake position) with respect to (i, k) if Type A n : k k i i + 2 and k k i i (mod 2) (respectively, min{2n + 2 i i, i + i } k k i i + 2 and k k i i (mod 2) The point (i, k ) is in minimal snake position to (i, k) if k k is equal to the given lower bound
11 Motivations Cluster algebras Representations of quantum affine algebras Snake modules Hernandez-Leclerc Conjecture: a certain subcategory C l of the category of all finite dimensional U q (ĝ)-modules has a cluster algebra structure In summer school, Leclerc introduced the conjecture and listed known results The fact that prime snake modules are prime was proved by Mukhin and Young
12 Our results We proved the following result Theorem Prime snake modules of type A n, B n are real Idea of the proof: we use "path description of q-characters" introduced by Mukhin and Young in 2012 Remark Prime snake modules of type A n are real also follows from a recent result of Lapid and Minguez in 2017 They proved (in the language of representations of p-adic groups) that a large family of U q (ĝ)-modules (containing prime snake modules) of type A n are real
13 Our main results are Theorem (1) Prime snake modules can be obtained using a sequence of mutations from the initial cluster of the cluster algebras constructed by Hernandez and Leclerc (2) We obtained a system of equations satisfied by q-characters of prime snake modules The system of equations contains all prime snake modules and only contains prime snake modules It is a natural generalization of T-systems I will show how to find mutation sequences
14 Mutation sequences and some examples For every finite dimensional Lie algebra g, Hernandez and Leclerc constructed a cluster algebra with an infinite quiver Q, for example, in type A 5, see Figure We denote = L(Y 4,aq 2Y 4,a ) Figure: Quiver A 5
15 How to get L(2 4 ), it is a non-minimal prime snake module, and L( )
16 [4 0 ][4 2 ] = [ ] + [3 1 ][ ] [ ][ ] = [4 2 ][ ] + [ ][5 3 ]
17 [ ][ ] = [ ][ ] + [ ][ ] [3 1 ][3 3 ] = [ ][ ] + [2 2 ][4 2 ]
18 [3 33 1][3 53 3] = [3 3][ ] + [2 42 2][4 44 2] [ ][ ] = [ ][ ] + [ ][ ]
19 [1 1][1 320] = [20][1 31 1] + [2 220], [1 31 1][ ] = [1 320][ ] + [ ], [ ][ ] = [ ][ ] + [ ] [2 220][2 420] = [ ][20] + [1 320][3 320], [ ][ ] = [2 420][ ] + [ ][ ]
20 [5 3 ][ ] = [ ][4 2 ] + [ ] [4 44 2][4 63 3] = [3 3][ ] + [5 54 2][3 53 3]
21 [ ][ ] = [ ][2 4 ] + [ ][ ]
22 S-systems We obtained a series of exchange relations in the process of mutations They form a system of equations which we called S-systems, which is a natural generalization of T -systems These equations in S-systems are of the form [S 1 ][S 2 ] = [S 3 ][S 4 ] + [S 5 ][S 6 ], where S i, i = 1,, 6, are certain prime snake modules For example: [L( )][L( )] = [L( )][L(2 4 )] + [L( )][L( )]
23 Thank you!
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