Quantum Plactic and Pseudo-Plactic
Algebras
Todor Popov
Abstract We review the Robinson–Schensted–Knuth correspondence in the light
of the quantum Schur–Weyl duality. The quantum plactic algebra is defined to be
a Schur functor mapping a tower of left modules of Hecke algebras into a tower of
U
q
gl-modules. The functions on the quantum group carry a U
q
gl-bimodule structure
whose combinatorial spirit emerges in the RSK algorithm. The bimodule structure
on the algebra of biletter words is used for a functorial formulation of the quantum
pseudo-plactic algebra. The latter algebra has been proposed by Daniel Krob and
Jean-Yves Thibon as a higher noncommutative analogue of the quantum torus.
1 Schur–Weyl Duality
Let us denote by S the tower of the symmetric groups S
r
, S =
r≥0
S
r
. The group
algebra C[S
r
] is the space of functions on S
r
. The algebra C[S
r
] is a left and right
module over itself. Similarly, the space of functions C[S] is a left and right module
on itself, provided that each C[S
r
] acts nontrivially only on its own level r .
The Hecke algebra H
r
(q) is a deformation of the group algebra C[S
r
]
T
i
T
i+1
T
i
= T
i+1
T
i
T
i+1
i = 1,..., r − 1
T
i
T
j
= T
j
T
i
|i − j |≥ 2
T
2
i
= 1 + (q − q
−1
)T
i
i = 1,..., r − 1.
(1)
The specialization of the formal parameter q to q = 1 yields the Coxeter relations
of the symmetric group S
r
generated by the elementary transpositions s
i
= (ii + 1).
For generic values of q one has an isomorphism of algebras H
r
(q)
∼
= C[S
r
].
T. Popov (B )
INRNE, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria
e-mail: tpopov@koc.edu.tr
T. Popov
Koç University, Rumeli Feneri Mh., 34450 Istanbul, Turkey
© Springer Nature Singapore Pte Ltd. 2016
V. Dobrev (ed.), Lie Theory and Its Applications in Physics,
Springer Proceedings in Mathematics & Statistics 191,
DOI 10.1007/978-981-10-2636-2_32
441