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Open research questions in Algebraic structures and combinatorial models

142 gap statements mined from Algebraic structures and combinatorial models papers in our 4.5M-paper local library, which holds 1,245 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

What the literature leaves open

  • The paper investigates the cohomology rings of restricted Lie algebras in characteristic two, which is a gap in the current literature.

    Restricted graph Lie algebras in characteristic two · 2026 · DOI

  • Future research could involve the study of braided Z2-crossed tensor categories for odd groups. Future research could involve the application of the construction to other areas of physics.

    Modular <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> -crossed Tambara-Yamagami-like categories for even groups · 2026 · DOI

  • The lack of a general method for constructing braided Z2-crossed tensor categories for even groups. The need for a physically motivated approach to construct these categories.

    Modular <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> -crossed Tambara-Yamagami-like categories for even groups · 2026 · DOI

  • Related with this, Lou et al. gave a definition of Poisson double extension - which may be considered as an analogue of double Ore extension -, and showed that algebras in a class of double Ore extensions are deformation quantizations of Poisson double extensions.

    A View Toward Homomorphisms and CV-Polynomials Between Double Ore Extensions · 2026 · DOI

  • The Bisognano-Wichmann property has not been established for non-unitary Wightman conformal field theories. The existing results are limited to unitary theories.

    The Bisognano-Wichmann Property for Non-unitary Wightman Conformal Field Theories · 2026 · DOI

  • The current understanding of the structural relations between coefficients is limited. The Laurent expansion is not in a compact and structurally explicit form.

    Structural identities, mode-level reformulation, and a three-term alternative for α−1 in the SO(3,3) matrix model on P3 · 2026 · DOI

  • The gap is the lack of understanding of the recursive structures of molecules and cells in Gelfand $S_n$-graphs. The paper identifies the need for a recursive structure of $S_n$ to discuss the action of $S_n$ on molecules.

    Recursive structures of molecules and cells in Gelfand $S_n$-graphs · 2026

  • The paper suggests further research on generalizing the result to arbitrary rectangles. The paper suggests further research on understanding the spin statistic more directly at the level of standard Young tableaux.

    Thrall's problem for two rows · 2026

  • The paper identifies a gap in the field of Lie algebras, namely the lack of a combinatorial description of the character of the higher Lie module. The paper also identifies a gap in the understanding of Thrall's problem.

    Thrall's problem for two rows · 2026

  • The category of representations of a conformal net A was not known to be a balanced W^*-tensor category. The result generalizes to the larger context of a group acting on A.

    The balanced structure on the category of representations of a conformal net · 2026

  • The paper identifies a need for a new method for computing Ext-groups of twisted functors.

    Quantum Troesch complex · 2026

  • The paper does not study all possible Argyres-Douglas theories. The method is limited to cases where the 3d TFT has a finite number of half-BPS monopole operators.

    3D TFTs and boundary VOAs from BPS spectra of (G, G′) Argyres-Douglas theories · 2026 · DOI

  • The irreducibility of certain $\widehat{\mathfrak{sl}}_2$-modules of Wakimoto type was not previously established. The paper fills this gap by demonstrating that these modules admit a Wakimoto-type realization at both critical and non-critical levels.

    Irreducibility of Certain $\widehat{\mathfrak{sl}}_2$-Modules of Wakimoto Type · 2026 · DOI

  • The lack of explicit and minimal generating sets for the centres in all classical types. The need for a new connection between higher-order quantum Casimir elements and hook partitions.

    Higher-order Casimir elements and hook partitions for quantum groups of types B, C and D · 2026 · DOI

  • The study suggests further exploration of the connection between braided finite automata and representation theory. The proposal can be extended to other quantum algebras. The study can be used as a basis for further research on the construction of canonical bases for irreducible representations of quantum algebras.

    Braided finite automata and representation theory · 2026 · DOI

  • The lack of a framework that accounts for wave-particle duality. The need for a demonstration of the structure at exact precision.

    Wave-Particle Duality from the Crystal Topos Terminal Structure: A Six-Receipt Demonstration that Wave Is Continuous Tomita–Takesaki Modular Flow, Particle Is the Discrete Krieger Ergodic Action on the Same Type III_0 Factor, the Tick σ_{2π} Is the Unit of Conversion, Decoherence Is Automatic Kronecker Scrambling at log 3 / log 2, and Wavefunction Collapse Is Forced Sector Commitment at the Cap Layer 43 · 2026 · DOI

  • A technical challenge is the complexity of the Auslander-Reiten quiver and exceptional tubes. Another challenge is the need to leverage prior work on Brauer graph algebras and representation theory.

    Universal deformation rings of a special class of modules over generalized Brauer tree algebras · 2026 · DOI

  • The gap is the lack of understanding of universal deformation rings for certain modules over generalized Brauer tree algebras. The paper identifies the need for a better understanding of these rings and their classification.

    Universal deformation rings of a special class of modules over generalized Brauer tree algebras · 2026 · DOI

  • Lemma 6.8 establishes isomorphisms ΨG(V) ≅ OΩ ⊗K V and ΨH(W) ≅ DDr(OPn−1 ⊗K W) for representations with trivial Gm action, but does not address the case where Gm acts non-trivially or provide a unified treatment of the relationship between Lubin-Tate bundles and Drinfeld bundles for higher-level representations.

    THE CATEGORIES OF LUBIN-TATE AND DRINFELD BUNDLES · 2026 · DOI

  • The functors ΨG and ΨH are defined as direct limits over levels m via Lm ⊗K −, but the paper does not characterize the image of these functors or determine which Lubin-Tate bundles in VectConG_LT(Ω) arise from smooth representations in Repsm_K(G) and Repsm_K(H).

    THE CATEGORIES OF LUBIN-TATE AND DRINFELD BUNDLES · 2026 · DOI

  • The results in Section 3 allow us to create a new algorithm for elementary abelian groups, so groups of the type (Z/p)k. Naturally, one might want to consider other p−groups or at least groups of the type (Z/pm)k for m > 1. In this section we will start in that direction by showing how we can use Nakayama’s Lemma to connect our elementary case to the (Z/pm)k case, as well as discuss some related reductions.

    An algorithm for invariants of elementary abelian groups · 2026

  • The paper suggests studying higher powers of the monodromy operator in the trace formula. The paper suggests investigating the 4d SCFT/2d VOA correspondence using other types of quantum field theories.

    3D TFTs and boundary VOAs from BPS spectra of (G, G′) Argyres-Douglas theories · 2026 · DOI

  • To apply the non-unitary version of the Bisognano-Wichmann property to the study of non-unitary conformal field theories. To extend the results to more general classes of conformal field theories.

    The Bisognano-Wichmann Property for Non-unitary Wightman Conformal Field Theories · 2026 · DOI

  • The paper identifies a gap in the understanding of the composition of reduction and inverse-reduction functors for standard modules of the affine vertex-operator algebra associated with $\mathfrak{sl}_2$.

    Reduction and inverse-reduction functors I: standard $\mathsf{V^k}(\mathfrak{sl}_2)$-modules · 2026

  • The paper identifies a gap in the construction of canonical bases for irreducible representations of quantum algebras. The study aims to propose a systematic means for constructing canonical bases.

    Braided finite automata and representation theory · 2026 · DOI

Most-cited papers in Algebraic structures and combinatorial models

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Open questions raised by papers listed on this page include: (1) Future research could involve the study of braided Z2-crossed tensor categories for odd groups (“Modular <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"…”, 2026); (2) The Bisognano-Wichmann property has not been established for non-unitary Wightman conformal field theories (“The Bisognano-Wichmann Property for Non-unitary Wightman Conformal Field…”, 2026); (3) The lack of explicit and minimal generating sets for the centres in all classical types (“Higher-order Casimir elements and hook partitions for quantum groups of types…”, 2026); (4) The lack of a framework that accounts for wave-particle duality (“Wave-Particle Duality from the Crystal Topos Terminal Structure: A Six-Receipt…”, 2026); (5) The gap is the lack of understanding of universal deformation rings for certain modules over generalized Brauer tree algebras (“Universal deformation rings of a special class of modules over generalized…”, 2026). 142 gap statements have been mined from Algebraic structures and combinatorial models papers in our 4.5M-paper local library, which holds 1,245 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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