Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Plethysm is in #BQP

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that every plethysm coefficient is the count of accepted witnesses of a polynomial-size quantum circuit — so plethysm is in #BQP and deciding positivity is in QMA.

desk verdict General plethysm in #BQP is a real advance, but the algorithm as written assumes an unconstructed basis-change isometry. read the letter →

arxiv 2602.08441 v1 pith:5YPZ6HIL submitted 2026-02-09 quant-ph cs.CCmath.COmath.RT

Matthias Christandl , Aram W. Harrow , Greta Panova , Pietro M. Posta , Michael Walter This is my paper
classification quant-phcs.CCmath.COmath.RT MSC 05E0505E1020C3068Q12 PACS 03.67.Ac
keywords plethysmcoefficients#BQPQMAbranchingmultiplicitiesSchurtransformSchur-WeyldualityKroneckerrepresentation-theoretic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's central claim is that a broad class of representation-theoretic multiplicities — branching multiplicities for products of general linear groups, including plethysm, Kronecker, Littlewood-Richardson, and Kostka coefficients — can be computed by quantum counting circuits, placing them in the class #BQP. For the plethysm coefficient a^λ_{μν}, the paper exhibits a quantum verifier whose acceptance subspace has dimension exactly that coefficient, so telling whether the coefficient is positive lands in QMA. A sympathetic reader should care because these numbers have resisted combinatorial interpretations for decades; the result shows that quantum computers could certify them even if no positive formula exists. The proof works by embedding the relevant representation into a tensor power using inverse Schur transforms and then performing strong Fourier sampling. If correct, it unifies previously known quantum-complexity results for Kronecker and special plethysm coefficients into one framework.

What carries the argument

The load-bearing object is the branching multiplicity for products of general linear groups: the multiplicity of an irreducible H-module in the restriction of an irreducible G-module along a succinctly specified homomorphism, specified by how the defining modules of G decompose over H. The argument is carried by the quantum Schur transform, the unitary change of basis from the computational basis of (C^d)^{⊗N} to the Schur basis indexed by a partition, a basis vector of the Weyl module, and a basis vector of the Specht module. The key mechanism is a two-step recipe: embed the representation to be decomposed into a model representation (a tensor power) via inverse Schur transforms, then stron

What would settle it

The claim is settled by whether the intermediate isotypic measurement can be implemented by a circuit of size polynomial in log d_i: exhibiting explicit such circuits for the exponentially large local dimensions would confirm the theorem, whereas a superpolynomial lower bound on that basis change for any family of instances would refute the claimed polynomial runtime.

Watch

Extended reading notes

Core claim

The central claim is constructive: the plethysm coefficient a^λ_{μν} equals the dimension of the accepting subspace of a polynomial-size quantum circuit, so computing plethysm coefficients is in #BQP. The circuit embeds the restricted module into (C^n)^{⊗|ν||μ|} via two layers of inverse Schur transforms, applies a Schur transform to the whole space, and accepts on measurement outcome (λ, p0). Every operation is equivariant, so the accepted subspace is exactly the λ-isotypic component. The same construction, with an intermediate isotypic measurement for defining modules that reduce nontrivially over H, handles arbitrary branching multiplicities for products of general linear groups; positivi

Load-bearing premise

The polynomial runtime rests on the assumption that the isotypic measurement in the general algorithm — which identifies the computational basis of each exponentially large defining module with the Fourier basis for H — can be performed efficiently; the paper asserts this basis identification without giving a circuit-level construction, and if implementing it costs more than poly(log d_i) the algorithm is no longer polynomial time.

Editorial extensions

If this is right

  • Computing plethysm coefficients is in #BQP: there is a polynomial-size quantum circuit whose accepted-witness subspace has dimension exactly a^λ_{μν}.
  • Deciding whether a plethysm coefficient is positive is in QMA, so a quantum verifier can certify nonzero coefficients.
  • The same holds for the general branching problem for products of general linear groups, subsuming Kronecker, Littlewood-Richardson, and Kostka coefficients.
  • Restriction coefficients of the symmetric group inside GL(n) are in #BQP as an immediate corollary.
  • If plethysm later turns out not to be in #P, the theorem would imply a separation between #P and #BQP, sharpening the stakes of the open combinatorial-interpretation problem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the embedding-plus-strong-Fourier-sampling recipe suggests a template for other multiplicity problems — such as fusion coefficients of compact Lie groups or multiplicities in tensor categories — whenever a model representation with efficient strong Fourier sampling exists.
  • Editorial extension: for the fermionic N-representability case (μ=(m), ν=(1^d)), a quantum verifier for the plethysm coefficient offers a route toward quantum-certifiable consistency of fermionic reduced density matrices.
  • Editorial extension: if the high-dimensional Schur transform can later be implemented exactly, the same framework would likely upgrade the QMA statement to a perfect-completeness version, since the circuit structure itself is a projective measurement.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proves that a broad class of representation-theoretic branching multiplicities for products of general linear groups, including all plethysm coefficients, is in #BQP, and consequently that positivity of these multiplicities lies in QMA. The quantum algorithm embeds the relevant Weyl module into a tensor product space via multiple layers of inverse Schur transforms, performs intermediate isotypic measurements, and concludes with strong Schur sampling; the dimension of the accepting subspace is shown to equal the desired multiplicity. The paper also gives a GapP containment for these multiplicities and fixed-parameter classical algorithms for plethysm coefficients when the number of rows of λ and the size of μ are fixed.

Significance. If correct, the central result resolves a notable open problem by showing that plethysm coefficients, whose membership in #P is unknown, are in #BQP. The approach unifies prior quantum algorithms for Kronecker and special plethysm coefficients and introduces a general framework (Appendix A) that captures both Schur-transform and generalized-phase-estimation methods. The paper also provides a GapP upper bound and classical fixed-parameter algorithms, which are of independent interest. The high-level construction is elegant, and the analysis of the accepted-subspace dimension is coherent. However, the proof of the #BQP claim currently omits a load-bearing circuit-level ingredient: the efficient implementation of the basis identification and the associated isometries.

major comments (2)
  1. [Section 5.2, Step 3; Definition 2.3] The algorithm assumes that the computational basis of each defining module C^{d_i} can be identified with the H-Fourier basis, making the isotypic measurement a standard basis measurement. However, the Schur transform circuit [8] used in Steps 2, 4, and 5 operates in the standard basis of C^{d_i}. If the computational basis is instead identified with the GT basis of the H-decomposition, the paper must provide a polynomial-size circuit for the basis change U_i (or for the ranking/unranking maps between GT labels and standard indices). No such construction is given. This gap is load-bearing for Theorem 5.1: without it, the isotypic measurement on an exponentially large qudit and the subsequent inverse Schur embeddings of single H-irreducible registers are not justified. The same issue affects Section 5.1, where Step 1 identifies C^{d_ν} with the GT basis of {ν}_n.
  2. [Section 5.1, Step 2] After Step 1, the state is in ({ν}_n)^{⊗|μ|}, where each register is a qudit of dimension dim{ν}_n. Step 2 applies an inverse Schur transform to each register to embed it into (C^n)^{⊗|ν|}. However, the inverse Schur transform circuit from [8] is a unitary on the full tensor product space (C^n)^{⊗|ν|}; it does not act on a single qudit. The algorithm needs an isometry V: {ν}_n → (C^n)^{⊗|ν|}, which is not supplied. The sentence 'by Schur–Weyl duality' plus adding ancillas does not specify how V is implemented coherently. Without a polynomial-size circuit for V, the complexity analysis of Theorem 5.1 does not establish membership in #BQP.
minor comments (4)
  1. [Section 5.1, Step 2] Typo: 'to each of the |λ| registers' should read 'to each of the |μ| registers', since Step 1 produces |μ| registers before the second embedding.
  2. [Section 2.1] The notation 'λ⊢ d n' is nonstandard and ambiguous; it should be written as λ ∈ P_d(n) or λ ⊢_d n.
  3. [Abstract] Several spacing issues in the abstract, e.g., 'in#BQP' and 'inGapP' should be 'in #BQP' and 'in GapP'.
  4. [Theorem 5.1 proof] In the complexity analysis, 'the Schur transforms in Steps 5 and 7 are transforms on H' should say 'on the representation spaces for H', since H is the group, not a Hilbert space.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central derivation is self-contained and rests on an external Schur-transform result.

full rationale

The paper's central claim (Theorems 1.2 and 1.3) is established by explicitly constructing a quantum verifier whose accepting subspace has dimension equal to the branching or plethysm multiplicity. These multiplicities are defined independently in Eq. (1.2) and Eq. (2.3), and the algorithm embeds the relevant modules into tensor powers via Schur-Weyl duality, applies inverse and forward Schur transforms, and performs strong Schur sampling. The input data in Definition 2.3 parameterize the homomorphism by the decomposition of the defining modules; Lemma 2.2 proves that this data determines the branching multiplicity, which is a genuine reduction rather than a definitional identity. The only load-bearing external ingredient, the high-dimensional quantum Schur transform with polylogarithmic dependence on local dimension, is cited to [8] (Burchardt et al., 2025), which is not authored by the present researchers. Self-citations ([16], [17], [26], [51]) appear only in background, comparison, or classical-algorithm sections and are not used to justify the #BQP membership claims. The reviewer-flagged concern that the computational-basis/Fourier-basis identification in Definition 2.3 and the intermediate isotypic measurement in Section 5.2, Step 3, lack an explicit circuit-level construction is an implementation-completeness risk, not a circularity: the target multiplicity is not assumed as an input, no fitted parameter is renamed as a prediction, and no cited result by the present authors is invoked to forbid alternatives or to carry the main proof. Therefore no circular step can be exhibited under the required evidentiary standard.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No fitted parameters. The argument rests on an external Schur transform theorem, standard representation theory, and one unproved algorithmic premise about the implementability of isotypic measurements on single exponentially large registers.

assumptions (3)
  • domain assumption The high-dimensional quantum Schur transform can be implemented with gate complexity O(n^4 polylog(n,d,epsilon^{-1})) for exponentially large local dimension d (Theorem 4.3 of Ref [8]).
    The entire #BQP inclusion depends on polynomial dependence on log d when d is exponentially large; this is cited from Burchardt et al. 2025 and not proved in this paper.
  • standard math Schur-Weyl duality for GL(d) x S_n and the hook-content dimension formula hold and can be used to embed irreducible modules into tensor powers.
    Used throughout Section 5 to construct the embeddings and to bound the exponential dimensions in the complexity analysis.
  • ad hoc to paper The computational basis of each defining module C^{d_i} can be chosen and operated as the Fourier basis for H, making the isotypic measurement in Section 5.2 Step 3 implementable in polynomial time.
    This is the load-bearing algorithmic premise of the branching algorithm; it is asserted via Definition 2.3 and Lemma 2.2, but no efficient circuit construction is given.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Plethysm is in #BQP." pith.science (2026). https://pith.science/paper/5YPZ6HIL

@misc{pith2026260208441,
  author       = {Pith},
  title        = {Pith review of: Plethysm is in #BQP},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5YPZ6HIL}},
  note         = {Machine review of arXiv:2602.08441}
}
read the original abstract

Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #P. Whether this holds more generally is considered an important open problem in mathematics and computer science, with relevance for geometric complexity theory and quantum information. Recent work has investigated the quantum complexity of particular multiplicities, such as the Kronecker coefficients and certain special cases of the plethysm coefficients. Here, we show that a broad class of representation-theoretic multiplicities is in #BQP. In particular, our result implies that the plethysm coefficients are in #BQP, which was only known in special cases. It also implies all known results on the quantum complexity of previously studied coefficients as special cases, unifying, simplifying, and extending prior work. We obtain our result by multiple applications of the Schur transform. Recent work has improved its dependence on the local dimension, which is crucial for our work. We further describe a general approach for showing that representation-theoretic multiplicities are in #BQP that captures our approach as well as the approaches of prior work. We complement the above by showing that the same multiplicities are also naturally in GapP and obtain polynomial-time classical algorithms when certain parameters are fixed.

Figures

Figures reproduced from arXiv: 2602.08441 by the authors.

Figure 1
Figure 1. Quantum circuit for the plethysm coefficient [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. A circuit representation of the algorithm for [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Quantum circuit to establish that a representation-theoretic multiplicity is in [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Left: Generalized phase estimation (GPE) circuit (with optional uncomputation [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Counting in logarithmic space

    math.CO 2026-07 conditional novelty 7.0 of 10

    Many known combinatorial and number-theoretic counting functions belong to the log-space counting class #L, with a new polylog-space verifier for GL2 plethysm coefficients.

Reference graph

Works this paper leans on

55 extracted references · 5 linked inside Pith · cited by 1 Pith paper

  1. [8]

    Burchardt, J

    A. Burchardt, J. Fei, D. Grinko, M. Larocca, M. Ozols, S. Timmerman, and V. Visnevskyi. High-dimensional quantum Schur transforms, 2025. arXiv:2509.22640 [quant-ph]. 6, 16

    arXiv 2025

  2. [1]

    Altunbulak and A

    M. Altunbulak and A. Klyachko. The Pauli principle revisited.Communications in Mathematical Physics, 282(1):287–322, 2008. 3, 4

    2008

  3. [2]

    Bacon, I

    D. Bacon, I. Chuang, and A. Harrow. Efficient quantum circuits for Schur and Clebsch-Gordan transforms. Physical Review Letters, 97(17):170502, Oct. 2006. 16

    2006

  4. [3]

    A. I. Barvinok. A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed.Mathematics of Operations Research, 19(4):769–779, 1994. 21

    1994

  5. [4]

    R. Beals. Quantum computation of Fourier transforms over symmetric groups. InProceedings of the 29th annual ACM symposium on Theory of Computing - STOC ’97, pages 48–53, El Paso, Texas, United States,

  6. [5]

    R. E. Borland and K. Dennis. The conditions on the one-matrix for three-body fermion wavefunctions with one-rank equal to six.Journal of Physics B: Atomic and Molecular Physics, 5(1):7–15, January 1972. 3

    1972

  7. [6]

    Bravyi, A

    S. Bravyi, A. Chowdhury, D. Gosset, V. Havlíček, and G. Zhu. Quantum complexity of the Kronecker coefficients.PRX Quantum, 5(1):010329, 2024. 2, 4, 5, 7, 25, 26, 27

    2024

  8. [7]

    M. Brion. Stable properties of plethysm: on two conjectures of Foulkes.Manuscripta Mathematica, 80:347–371,

Show all 55 references
  1. [9]

    Bürgisser, M

    P. Bürgisser, M. Christandl, and C. Ikenmeyer. Even partitions in plethysms.Journal of Algebra, 328(1):322– 329, Feb. 2011. 3

    2011

  2. [10]

    Bürgisser, M

    P. Bürgisser, M. Christandl, K. Mulmuley, and M. Walter. Membership in moment polytopes is in NP and coNP.SIAM Journal on Computing, 46(3):972–991, 2017. 2

    2017

  3. [11]

    Bürgisser and C

    P. Bürgisser and C. Ikenmeyer. The complexity of computing Kronecker coefficients.FPSAC’08 - 20th International Conference on Formal Power Series and Algebraic Combinatorics, DMTCS Proceedings vol. AJ, 01 2008. 5, 21 27

    2008

  4. [12]

    Bürgisser and C

    P. Bürgisser and C. Ikenmeyer. Geometric complexity theory and tensor rank. InProceedings of the Forty- Third Annual ACM Symposium on Theory of Computing (STOC ’11), pages 509–518, New York, NY, USA,

  5. [13]

    Bürgisser and C

    P. Bürgisser and C. Ikenmeyer. Deciding positivity of Littlewood–Richardson coefficients.SIAM Journal on Discrete Mathematics, 27(4):1639–1681, 2013. 2

    2013

  6. [14]

    Bürgisser, C

    P. Bürgisser, C. Ikenmeyer, and J. Hüttenhain. Permanent versus determinant: not via saturations.Proceedings of the American Mathematical Society, 145(3):1247–1258, 2017. 3

    2017

  7. [15]

    Bürgisser, C

    P. Bürgisser, C. Ikenmeyer, and G. Panova. No occurrence obstructions in geometric complexity theory. Journal of the American Mathematical Society, 32(1):163–193, 2019. 3

    2019

  8. [16]

    Christandl, B

    M. Christandl, B. Doran, and M. Walter. Computing multiplicities of Lie group representations. In2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 639–648. IEEE, 2012. 2, 4, 21

    2012

  9. [17]

    Christandl, A

    M. Christandl, A. Harrow, and M. Walter. Presentation at the Algorithms and Complexity in Algebraic Geometry Reunion Workshop. Presented by M. Walter at the Simons Institute for the Theory of Computing, Berkeley, Dec. 2015. 2, 25

    2015

  10. [18]

    Christandl, A

    M. Christandl, A. W. Harrow, and G. Mitchison. On nonzero Kronecker coefficients and what they tell us about spectra.Communications in Mathematical Physics, 270(3):575–585, 2007. 3

    2007

  11. [19]

    Christandl and G

    M. Christandl and G. Mitchison. The spectra of quantum states and the Kronecker coefficients of the symmetric group.Communications in Mathematical Physics, 261(3):789–797, 2006. 3

    2006

  12. [20]

    A. J. Coleman and V. I. Yukalov.Reduced Density Matrices: Coulson ’s Challenge. Lecture Notes in Chemistry, Vol. 72. Springer Berlin Heidelberg, 2000. 3

    2000

  13. [21]

    Colmenarejo, R

    L. Colmenarejo, R. Orellana, F. Saliola, A. Schilling, and M. Zabrocki. The mystery of plethysm coefficients. Open Problems in Algebraic Combinatorics, 110:275–292, 2024. 2

    2024

  14. [22]

    de Boeck, R

    M. de Boeck, R. Paget, and M. Wildon. Plethysms of symmetric functions and highest weight representations. Transactions of the American Mathematical Society, 374(11):8013–8043, 2021. 2

    2021

  15. [23]

    Dörfler, C

    J. Dörfler, C. Ikenmeyer, and G. Panova. On geometric complexity theory: Multiplicity obstructions are stronger than occurrence obstructions.SIAM Journal on Applied Algebra and Geometry, 4(2):354–376, 2020. 2, 3

    2020

  16. [24]

    Fischer and C

    N. Fischer and C. Ikenmeyer. The computational complexity of plethysm coefficients.Computational Complexity, 29:1–43, 2020. 2, 21, 22, 23

    2020

  17. [25]

    H. Foulkes. Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form. Journal of the London Mathematical Society, 1(3):205–209, 1950. 4

    1950

  18. [26]

    A. W. Harrow. Applications of coherent classical communication and the Schur transform to quantum information theory, 2005. PhD thesis. 6, 7, 16, 26

    2005

  19. [27]

    Ikenmeyer, K

    C. Ikenmeyer, K. Mulmuley, and M. Walter. On vanishing of Kronecker coefficients.Computational Complexity, 26:949–992, 2017. 2

    2017

  20. [28]

    Ikenmeyer and G

    C. Ikenmeyer and G. Panova. Rectangular Kronecker coefficients and plethysms in geometric complexity theory.Advances in Mathematics, 319:40–66, 2017. 3

    2017

  21. [29]

    Ikenmeyer and S

    C. Ikenmeyer and S. Subramanian. A remark on the quantum complexity of the Kronecker coefficients. arXiv:2307.02389, 2023. 2, 4, 5, 7, 25, 26, 27

    2023 arXiv

  22. [30]

    James and A

    G. James and A. Kerber.The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1984. 7

    1984

  23. [31]

    Kahle and M

    T. Kahle and M. Michałek. Plethysm and lattice point counting.Foundations of Computational Mathematics, 16(5):1241–1261, 2016. 2

    2016

  24. [32]

    A. A. Klyachko. Quantum marginal problem and representations of the symmetric group.arXiv e-print quant-ph/0409113, 2004. 3

    2004 arXiv

  25. [33]

    A. A. Klyachko. Quantum marginal problem and N-representability.arXiv e-print quant-ph/0511102, 2005. 4

    2005 arXiv

  26. [34]

    Knutson and T

    A. Knutson and T. Tao. The honeycomb model ofGLn(C )tensor products I: Proof of the saturation conjecture.Journal of the American Mathematical Society, 12(4):1055–1090, 1999. 2

    1999

  27. [35]

    H. Krovi. An efficient high dimensional quantum Schur transform.Quantum, 3:122, Feb. 2019. 16

    2019

  28. [36]

    Larocca and V

    M. Larocca and V. Havlicek. Quantum algorithms for representation-theoretic multiplicities.arXiv:2407.17649,

    arXiv

  29. [37]

    D. E. Littlewood. Polynomial concomitants and invariant matrices.Journal of the London Mathematical Society, s1-11(1):49–55, 1936. 1

    1936

  30. [38]

    D. E. Littlewood. Products and plethysms of characters with orthogonal, symplectic and symmetric groups. Canadian Journal of Mathematics, 10:17–32, 1958. 1, 8 28

    1958

  31. [39]

    D. E. Littlewood and A. R. Richardson. Group characters and algebra.Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 233(721-730):99–141,

  32. [40]

    I. G. Macdonald.Symmetric functions and Hall polynomials. Oxford University Press, 2. edition, 1998. 8

    1998

  33. [41]

    Mulmuley, H

    K. Mulmuley, H. Narayanan, and M. Sohoni. Geometric complexity theory III: on deciding nonvanishing of a Littlewood–Richardson coefficient.Journal of Algebraic Combinatorics, 36(1):103–110, 2012. 2

    2012

  34. [42]

    F. D. Murnaghan. The analysis of the Kronecker product of irreducible representations of the symmetric group.American Journal of Mathematics, 60(3):761–784, 1938. 1

    1938

  35. [43]

    Orellana, F

    R. Orellana, F. Saliola, A. Schilling, and M. Zabrocki. Plethysm and the algebra of uniform block permutations. Algebraic Combinatorics, 5(5):1165–1203, 2022. 2

    2022

  36. [44]

    Orellana, F

    R. Orellana, F. Saliola, A. Schilling, and M. Zabrocki. From quasi-symmetric to Schur expansions with applications to symmetric chain decompositions and plethysm.Electron. J. Combin., 31(4), 2024. 2

    2024

  37. [45]

    I. Pak. What is a combinatorial interpretation? InOpen Problems in Algebraic Combinatorics, volume 110 of Proc. Sympos. Pure Math., pages 191–260. Amer. Math. Soc., Providence, RI, 2024. 2, 5

    2024

  38. [46]

    Pak and G

    I. Pak and G. Panova. On the complexity of computing Kronecker coefficients.Comput. Complexity, 26:1–36,

  39. [47]

    I. Pak, G. Panova, and J. P. Swanson. A combinatorial interpretation for certain plethysm and Kronecker coefficients, 2025. 2, 22

    2025

  40. [48]

    Pak and C

    I. Pak and C. Robichaux. Signed combinatorial interpretations in algebraic combinatorics.arXiv:2406.13902,

    arXiv

  41. [49]

    G. Panova. Computational complexity in algebraic combinatorics. InCurrent Developments in Mathematics. Harvard University, International Press Boston, 2023. 2, 5

    2023

  42. [50]

    G. Panova. Complexity and asymptotics of structure constants. InOpen Problems in Algebraic Combinatorics, volume 110 ofProc. Sympos. Pure Math., pages 61–86. Amer. Math. Soc., Providence, RI, 2024. 2

    2024

  43. [51]

    G. Panova. Polynomial time classical versus quantum algorithms for representation theoretic multiplicities. arXiv preprint arXiv:2502.20253, 2025. 2, 5, 21, 22

    2025

  44. [52]

    B. E. Sagan.The symmetric group: representations, combinatorial algorithms, and symmetric functions, volume 203. Springer, 2013. 7

    2013

  45. [53]

    Stanley.Enumerative Combinatorics, volume 2

    R. Stanley.Enumerative Combinatorics, volume 2. Cambridge University Press, 1 edition, 1999. 7, 8

    1999

  46. [54]

    R. P. Stanley. Positivity problems and conjectures in algebraic combinatorics.Mathematics: Frontiers and Perspectives, pages 295–319, 2000. 1, 4 29

    2000

  47. [2011]

    Association for Computing Machinery. 3

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.