The two bosonizations of the CKP hierarchy: overview and character identities

CKP hierarchy and free Bosons

“…In this section, we will review the construction of CKP hierarchy by using free Bosons. Firstly, note that CKP hierarchy is corresponding to the infinite dimensional Lie algebra c ∞ = c∞ ⊕Cc [3,11,12,39], where c∞ = (a i,j ) i,j∈Z |a i,j = (−1) i+j+1 a −j+1,−i+1 , a i,j = 0, for |i − j| ≫ 0 .…”

“…However because of the square root relation in (1), the CKP hierarchy can not be expressed as the usual Hirota bilinear equation by using τ CW . After the works above, Anguelova [2,3] introduced a second bosonization of the CKP hierarchy by using an untwisted Heisenberg algebra generated by But there is no corresponding relations between wave functions and the tau functions in [2,3], since it seems difficult to determine that relation by noticing the corresponding Bosonization formulas.…”

“…, µ) and ξ( ∂, µ −1 ), the key is to know t, µ)A(t, −λ) . In fact, this can be seen from AB, CD] = A[B, C] + D − AC[B, D] + + [A, C] + DB − C[A, D] + B with [A, B] + = AB + BA, c (λ, µ) =X c (λ, e ξ(t,µ)−ξ(t,λ) e ξ( ∂,µ −1 )−ξ( ∂,λ −1 ) × (2j − 1) k>0 (−µ) j−1 )(−λ) −k − λ j−1 µ −k ∂ ∂t (−µ) j−terms depending on t odd .So the result of ∂ * λµ τ (2j−1,1) (t) can be derived by(3).Remark: Notice that the Adler-Shiota-van Moerbeke-Dickey formula in the CKP hierarchy can be obtained by∂ * λµ w(t, z) = w(t, z) e ξ( ∂,z −1 ) 1 2 ∂ * λµ τ 0 (t) + 2 ν∈1+2Z, ν>1 ∂ * λµ τ (ν,1) (t)z − ν+12 τ 0 (t) + 2 ν∈1+2Z, ν>1 τ (ν,1) (t)z − ν+1 (t) τ 0 (t) .…”

Quantum $W_{1+\infty}$ subalgebras of BCD type and symmetric polynomials

“…The BKP and DKP hierarchies have polynomial tau functions given by the Q-Schur polynomials [5,6], a specialization of the Hall-Littlewood polynomials at t = −1 [7,8]. 1 On the other hand, it was shown in [9] that the CKP hierarchy does not admit any polynomial tau function. Yet, symmetric polynomials associated to the Ĉ∞ algebra have been introduced recently by van de Leur, Orlov and Shiota in [10], and we will review their construction below.…”