Abstract
We introduce a new set of q-difference operators acting as raising operators on a family of symmetric polynomials which are characters of graded tensor products of current algebra g[ u] KR-modules [FL1] for g= A r . These operators are generalizations of the Kirillov-Noumi [KN] Macdonald raising operators, in the dual q-Whittaker limit t → ∞. They form a representation of the quantum Q-system of type A, see [DFK3]. This system is a subalgebra of a quantum cluster algebra, and is also a discrete integrable system whose conserved quantities, analogous to the Casimirs of U q (sl r + 1 ) , act as difference operators on the above family of symmetric polynomials. The characters in the special case of products of fundamental modules are class I q-Whittaker functions, or characters of level-1 Demazure modules or Weyl modules. The action of the conserved quantities on these characters gives the difference quantum Toda equations [E]. For other modules, the action of the conserved quantities is thus a generalization of these difference equations in the case of an arbitrary (higher level) tensor products of KR-modules.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 391-424 |
| Number of pages | 34 |
| Journal | Transformation Groups |
| Volume | 23 |
| Issue number | 2 |
| DOIs | |
| State | Published - Jun 1 2018 |
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ASJC Scopus subject areas
- Algebra and Number Theory
- Geometry and Topology
Cite this
DIFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA. / Di Francesco, P.; Kedem, R.
In: Transformation Groups, Vol. 23, No. 2, 01.06.2018, p. 391-424.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - DIFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA
AU - Di Francesco, P.
AU - Kedem, R.
PY - 2018/6/1
Y1 - 2018/6/1
N2 - We introduce a new set of q-difference operators acting as raising operators on a family of symmetric polynomials which are characters of graded tensor products of current algebra g[ u] KR-modules [FL1] for g= A r . These operators are generalizations of the Kirillov-Noumi [KN] Macdonald raising operators, in the dual q-Whittaker limit t → ∞. They form a representation of the quantum Q-system of type A, see [DFK3]. This system is a subalgebra of a quantum cluster algebra, and is also a discrete integrable system whose conserved quantities, analogous to the Casimirs of U q (sl r + 1 ) , act as difference operators on the above family of symmetric polynomials. The characters in the special case of products of fundamental modules are class I q-Whittaker functions, or characters of level-1 Demazure modules or Weyl modules. The action of the conserved quantities on these characters gives the difference quantum Toda equations [E]. For other modules, the action of the conserved quantities is thus a generalization of these difference equations in the case of an arbitrary (higher level) tensor products of KR-modules.
AB - We introduce a new set of q-difference operators acting as raising operators on a family of symmetric polynomials which are characters of graded tensor products of current algebra g[ u] KR-modules [FL1] for g= A r . These operators are generalizations of the Kirillov-Noumi [KN] Macdonald raising operators, in the dual q-Whittaker limit t → ∞. They form a representation of the quantum Q-system of type A, see [DFK3]. This system is a subalgebra of a quantum cluster algebra, and is also a discrete integrable system whose conserved quantities, analogous to the Casimirs of U q (sl r + 1 ) , act as difference operators on the above family of symmetric polynomials. The characters in the special case of products of fundamental modules are class I q-Whittaker functions, or characters of level-1 Demazure modules or Weyl modules. The action of the conserved quantities on these characters gives the difference quantum Toda equations [E]. For other modules, the action of the conserved quantities is thus a generalization of these difference equations in the case of an arbitrary (higher level) tensor products of KR-modules.
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U2 - 10.1007/s00031-018-9480-y
DO - 10.1007/s00031-018-9480-y
M3 - Article
AN - SCOPUS:85044932637
VL - 23
SP - 391
EP - 424
JO - Transformation Groups
JF - Transformation Groups
SN - 1083-4362
IS - 2
ER -