--- title: Quantum Corner Polynomials url: https://www.emergentmind.com/topics/quantum-corner-polynomials type: topic --- # Quantum Corner Polynomials Quantum corner polynomials are a family of partially symmetric polynomials introduced as a generalization of the Sergeev–Veselov super Macdonald polynomials and identified as the polynomial objects naturally attached to quantum corner vertex operator algebras (VOAs) \(q\widetilde{Y}_{M,L,N}\) [2508.12267]. Their defining feature is a three-sector structure—ordinary, super, and hyper variables—together with a combinatorial expansion over reverse semi-standard Young tritableaux. In the hierarchy described by the literature, they extend the earlier correspondence in which a quantum corner VOA \(q\widetilde{Y}_{L,0,N}[\Psi]\) produces Sergeev–Veselov super Macdonald polynomials after a specific specialization and limiting procedure [2504.17326]. ## 1. Definition and basic structure The central definition is combinatorial. For a partition \(\lambda\), the quantum corner polynomial is \[ QC_\lambda(x_1,\dots,x_N;x_{N+1},\dots,x_{N+M};x_{N+M+1},\dots,x_{N+M+L};q,t) = \sum_{T \in \operatorname{RSSYTT}(N,M,L;\lambda)} R_T(q,t)\,x_T . \] Here \(T\) runs over reverse semi-standard Young tritableaux of shape \(\lambda\) and type \((N,M,L)\), while \(x_T:=x_{i_1}\cdots x_{i_k}\) is the monomial determined by the tableau entries \((i_1,\dots,i_k)\). The variables are grouped into ordinary variables \(x_1,\dots,x_N\), super variables \(x_{N+1},\dots,x_{N+M}\), and hyper variables \(x_{N+M+1},\dots,x_{N+M+L}\), often renamed \(y_i\) and \(w_j\) for the latter two blocks. The coefficient \(R_T(q,t)\) is given as a product of tableau-dependent factors, including \(A_2(T;q,t)\), \(\psi_{T'_1}(t,q)\), \(\psi_{T_0}(q,t)\), hook-type factors \(H(\cdot)\), and a factor \(\left(\frac{q^{-1}t-1}{t^{-1}-1}\right)^{|T_2|}\) [2508.12267]. This definition places quantum corner polynomials within the Macdonald-theoretic tradition of weighted tableau sums, but with a variable set partitioned into three distinguished sectors. A plausible implication is that the combinatorics is designed not merely to deform symmetric polynomials, but to encode the additional boundary or corner data present in the corresponding VOA construction. ## 2. Tritableaux combinatorics The combinatorial indexing set is the set of reverse semi-standard Young tritableaux (RSSYTTs). A tableau \(T\) of type \((N,M,L)\) is filled with entries from \(\{1,\dots,N+M+L\}\), split into ordinary numbers \(1,\dots,N\), super numbers \(N+1,\dots,N+M\), and hyper numbers \(N+M+1,\dots,N+M+L\). The defining conditions are: 1. rows weakly decrease left to right, 2. columns weakly decrease top to bottom, 3. ordinary numbers strictly decrease down columns, 4. super numbers strictly decrease along rows. These are presented as the “partial super/hyper” analogue of the reverse semistandard conditions used in super Macdonald theory [2508.12267]. | Sector | Entries | Role in the polynomial | |---|---|---| | Ordinary | \(1,\dots,N\) | Variables \(x_1,\dots,x_N\) | | Super | \(N+1,\dots,N+M\) | Variables \(x_{N+1},\dots,x_{N+M}\), often \(y_i\) | | Hyper | \(N+M+1,\dots,N+M+L\) | Variables \(x_{N+M+1},\dots,x_{N+M+L}\), often \(w_j\) | The tritableau formalism is the basic combinatorial innovation distinguishing quantum corner polynomials from the two-sector super Macdonald setting. In this sense, the introduction of hyper numbers is the essential new ingredient: when the hyper sector is absent, the theory collapses to the previously known super case. ## 3. Partial symmetry The relevant symmetry notion is not full symmetry in all variables, but partial symmetry with respect to a fixed partition of the variable set into three blocks, \[ I_1=\{1,\dots,N\},\qquad I_2=\{N+1,\dots,N+M\},\qquad I_3=\{N+M+1,\dots,N+M+L\}. \] A polynomial \(g(x_1,\dots,x_n)\) is partially symmetric with respect to \(I_1,\dots,I_\ell\) if it is invariant under any permutation \(\sigma\) satisfying \(\sigma(I_i)=I_i\) for each block. For quantum corner polynomials, this means symmetry separately in the ordinary variables \(x_1,\dots,x_N\), in the super variables \(x_{N+1},\dots,x_{N+M}\), and in the hyper variables \(x_{N+M+1},\dots,x_{N+M+L}\), but not necessarily under permutations that mix these blocks. The paper’s main theorem on this point states that \[ QC_\lambda(x_1,\dots,x_N;x_{N+1},\dots,x_{N+M};x_{N+M+1},\dots,x_{N+M+L};q,t) \] is partially symmetric with respect to those three blocks [2508.12267]. This blockwise invariance is central to the concept. It distinguishes quantum corner polynomials from ordinary Macdonald polynomials, which are fully symmetric, and it also extends the super Macdonald situation by replacing a two-block symmetry pattern with a three-block one. ## 4. Relation to Sergeev–Veselov super Macdonald polynomials The parameter \(L\), which counts the hyper numbers, governs the reduction to the previously known super theory. When \(L=0\), the tritableau contains only ordinary and super entries, and the quantum corner polynomial reduces exactly to the Sergeev–Veselov super Macdonald polynomial: \[ QC_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t)=SP_\lambda(x_1,\dots,x_N;y_1,\dots,y_M;q,t) \quad \text{when } L=0. \] The paper presents this as a proposition and summarizes the hierarchy as \(L=0\) corresponding to super Macdonald polynomials and \(L>0\) corresponding to genuinely new quantum corner polynomials [2508.12267]. The immediate precursor is the earlier result for the quantum corner VOA \(q\widetilde{Y}_{L,0,N}[\Psi]\), which was described as a generalization of the quantum \(W_N\) algebra. In that setting, the vacuum current correlators \[ \langle 0|\widetilde{T}_{i_1}(z_1)\cdots \widetilde{T}_{i_k}(z_k)|0\rangle \] were shown, after specialization of the parameters \((q_1,q_2,q_3)=(q,q^{-1}t,t^{-1})\), application of the substitution map \(\mathfrak{F}_{\lambda,\xi}\), and the limit \(\xi\to t^{-1}\), to coincide with Sergeev–Veselov super Macdonald polynomials. In the notation of that paper, the ordinary corner VOA satisfies \(Y_{0,0,N}=W_N\), and the main theorem identifies the specialized correlator with \[ SP_\lambda\!\left( u_1,\dots,u_N;\; q^{\lambda_1}u_{N+1},\dots,q^{\lambda_M}u_{N+M};\; q,t \right) \] for \(\lambda\in H_{N,M}\), where \(H_{N,M}=\{\lambda\in\mathrm{Par}\mid \lambda_{N+1}\le M\}\) [2504.17326]. Quantum corner polynomials therefore occupy the next stage in the same lineage: they extend the corner/super Macdonald correspondence from a two-sector to a three-sector setting. ## 5. Correspondence with the quantum corner VOA The paper identifies quantum corner polynomials as the partially symmetric polynomials corresponding to the quantum corner VOA \(q\widetilde{Y}_{M,L,N}\). The VOA is built from the quantum toroidal \(\mathfrak{gl}_1\) algebra and its horizontal Fock representations. Its generating currents are denoted \(\widetilde{T}^{\vec c,\vec u}_m(z)\), with \(\vec c=(3^N1^M2^L)\), and these currents satisfy quadratic relations generalizing those of the quantum \(W_N\) algebra [2508.12267]. The main correspondence theorem states that a normalized vacuum matrix element of products of the VOA currents, after a limiting procedure and a specialization map \(\dualmap\), becomes exactly the quantum corner polynomial: \[ \lim_{\xi\to t^{-1} \left( \text{specialized } N_\lambda(z_1,\dots,z_k) \cdot \prod_{1\le i