arxiv: 2605.08085 · v1 · submitted 2026-05-08 · ✦ hep-th

Recognition: no theorem link

Fermionic trace relations and supersymmetric indices at finite N

Giorgos Eleftheriou , Ziming Ji , Sameer Murthy

Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:51 UTC · model grok-4.3

classification ✦ hep-th

keywords fermionic trace relationssupersymmetric indicesN=4 SYMfinite NGrassmann matricesU(N) invariantsmatrix modelstrace relations

0 comments

The pith

A 1/4-BPS supersymmetric index in N=4 SYM theory is independent of the rank N.

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

The paper examines invariants of U(N) matrices that include Grassmann-valued fermionic entries, weighted by fermion number, as they arise in supersymmetric indices of gauge theories. Grassmann matrices obey a nilpotency condition that forces their 2N-th power to vanish, generating extra trace relations not present in purely bosonic models. These fermionic relations can offset the usual bosonic ones so that the index grows or stays constant as N drops, rather than shrinking. For the minimal model with one fermion and one derivative that reproduces the 1/4-BPS index of N=4 SYM, the opposing relations cancel exactly and the index becomes independent of N. The authors prove this independence analytically and document the cancellations numerically.

Core claim

For the simple model involving one fermion and one derivative that corresponds to a 1/4-BPS supersymmetric index in N=4 SYM theory, the index is independent of N. This rank-independence is proved analytically and arises from exact cancellations between bosonic and fermionic trace relations.

What carries the argument

Cancellations between bosonic trace relations and the new fermionic trace relations generated by the vanishing of the 2N-th power of a Grassmann matrix.

If this is right

The value of the index computed at large N continues to hold at every finite N.
Analogues of the polarized Cayley-Hamilton identities and the Second Fundamental Theorem of invariants exist for models containing Grassmann matrices.
Smooth and singular limits of the most general supersymmetric index in N=4 SYM exhibit characteristic patterns when N is varied.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar cancellations could simplify finite-N calculations in other supersymmetric models that include fermions.
The observed rank-independence may extend to higher-BPS sectors or different gauge groups, offering a route to closed-form expressions at any N.
If the conjectured algebraic structures hold, they would furnish a systematic way to enumerate invariants in mixed bosonic-fermionic matrix models.

Load-bearing premise

The cancellations between bosonic and fermionic trace relations are exact and complete in the one-fermion, one-derivative model.

What would settle it

Explicit computation of the index for two different finite values of N, such as N=2 and N=3, that yields unequal results.

read the original abstract

We study invariants of bosonic and fermionic (Grassmann-valued) matrices under the adjoint action of $U(N)$, weighted by the fermion number. Such models naturally appear as the supersymmetric indices of supersymmetric gauge theories and are captured by $U(N)$ matrix models. We discuss two features of the fermionic models that are qualitatively different from bosonic models. Firstly, the $2N^\text{th}$ power of a Grassmann matrix vanishes, which gives rise to many new trace relations. Secondly, trace relations in models involving fermions could cause an increase in the supersymmetric index as $N$ decreases, in contrast with purely bosonic models. We focus on a simple model involving one fermion and one derivative that corresponds to a $\frac14$-BPS supersymmetric index in $\mathcal{N}=4$ SYM theory, in which we find that the index is independent of $N$. We prove this rank-independence analytically, and experimentally study the cancellations between bosonic and fermionic trace relations that lead to it. Based on these observations, we make some conjectures on resulting algebraic structures, including the analogue of the polarized Cayley-Hamilton identities and the Second Fundamental Theorem of invariants in the presence of Grassmann matrices. Finally, we present various (smooth and singular) limits of the most general supersymmetric index in $\mathcal{N}=4$ SYM theory, and study some patterns in their behavior as a function of $N$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves that a simple one-fermion model for the 1/4-BPS index in N=4 SYM is exactly independent of N through cancellations between bosonic and new fermionic trace relations.

read the letter

The key point is that this work proves N-independence for the supersymmetric index in a minimal model with one fermion and one derivative. The proof works by showing that every bosonic trace relation is cancelled exactly by a fermionic one that follows from the nilpotency condition on the Grassmann matrix, so the generating function ends up the same for all N. They also check this explicitly for small N to see the pattern of cancellations in action. This is new relative to the usual bosonic matrix-model literature on indices, where N-dependence typically decreases as rank drops. The conjectures on mixed bosonic-fermionic versions of the polarized Cayley-Hamilton identities and the second fundamental theorem are a reasonable next step from the observations and could be worth testing further. The paper is limited in scope: the rigorous result holds only for this simple case, while the sections on limits of the most general N=4 SYM index are mainly observational and do not deliver comparable proofs or a full classification. The algebraic claims rest on the explicit cancellations rather than fitting or circular definitions, and the abstract plus stress-test description give no sign of internal contradictions. This is aimed at people working on finite-N effects in supersymmetric gauge theories and AdS/CFT. It has a concrete, checkable central result plus some open algebraic questions, so it deserves a serious referee.

Referee Report

0 major / 2 minor

Summary. The manuscript examines invariants of bosonic and fermionic (Grassmann-valued) matrices under the U(N) adjoint action, weighted by fermion number, in the context of supersymmetric indices for gauge theories. It identifies qualitative differences from bosonic models, including new trace relations from the nilpotency condition that the 2N-th power of a Grassmann matrix vanishes. For a simple one-fermion, one-derivative model corresponding to the 1/4-BPS index in N=4 SYM, the authors analytically prove that the index is independent of N through exact cancellations between bosonic and fermionic trace relations. They provide experimental verification of these cancellations for small N, conjecture on algebraic structures such as polarized Cayley-Hamilton identities and the Second Fundamental Theorem for Grassmann matrices, and analyze smooth and singular limits of the general N=4 SYM supersymmetric index as a function of N.

Significance. If the central analytical proof holds, the result is significant because it establishes an exact rank-independence for a supersymmetric index, in contrast to bosonic models where the index generally decreases with decreasing N. The demonstration that fermionic trace relations precisely cancel bosonic ones to yield an N-independent generating function provides a concrete algebraic mechanism with potential implications for finite-N effects in N=4 SYM and related matrix models. The conjectures on generalized invariant theory for Grassmann variables and the study of index limits offer new directions for research in supersymmetric gauge theory and algebraic combinatorics.

minor comments (2)

[Abstract] The abstract and introduction would benefit from an explicit statement of the precise scope of the analytical proof (e.g., which classes of trace relations are included and whether the result extends immediately to multi-fermion models).
In the experimental section, the description of the numerical checks for small N should specify the range of N values tested and the method used to enumerate independent traces, to allow readers to assess the completeness of the observed cancellation pattern.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and positive summary of our manuscript, for highlighting the significance of the N-independence result, and for recommending minor revision. No specific major comments were raised in the report.

read point-by-point responses

Referee: The manuscript examines invariants of bosonic and fermionic (Grassmann-valued) matrices under the U(N) adjoint action, weighted by fermion number, in the context of supersymmetric indices for gauge theories. It identifies qualitative differences from bosonic models, including new trace relations from the nilpotency condition that the 2N-th power of a Grassmann matrix vanishes. For a simple one-fermion, one-derivative model corresponding to the 1/4-BPS index in N=4 SYM, the authors analytically prove that the index is independent of N through exact cancellations between bosonic and fermionic trace relations. They provide experimental verification of these cancellations for small N, conjecture on algebraic structures such as polarized Cayley-Hamilton identities and the Second Fundamental Theorem for Grassmann matrices, and analyze smooth and singular limits of the general N=4 SYM supersy

Authors: We thank the referee for this accurate summary of the manuscript. The description correctly captures the main results, including the analytical proof of N-independence for the one-fermion model and the conjectures on algebraic structures for Grassmann matrices. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives the N-independence of the 1/4-BPS index for the one-fermion one-derivative model via an explicit analytical proof: every bosonic trace relation is cancelled by a corresponding fermionic relation generated by the nilpotency condition (the 2N-th power of the Grassmann matrix vanishes), so the generating function reduces to the same expression independent of N. This follows directly from the algebraic properties of the matrices and the adjoint action, without any fitted parameters, self-definitional loops, or load-bearing self-citations. The experimental checks for small N are presented only as confirmation of the cancellations, not as input to the proof. The central claim is therefore a self-contained mathematical statement derived from the model's defining relations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of Grassmann algebras and the established correspondence between the chosen matrix model and the 1/4-BPS index; no free parameters are fitted and no new entities are postulated.

axioms (2)

standard math An N by N Grassmann matrix satisfies M^{2N} = 0.
Standard nilpotency property of fermionic matrices invoked to generate new trace relations.
domain assumption The one-fermion one-derivative matrix model corresponds to the 1/4-BPS supersymmetric index of N=4 SYM.
Stated directly in the abstract as the physical interpretation of the mathematical model.

pith-pipeline@v0.9.0 · 5565 in / 1537 out tokens · 50918 ms · 2026-05-11T01:51:22.668599+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

80 extracted references · 80 canonical work pages · 1 internal anchor

[1]

A. Y. Morozov,Unitary Integrals and Related Matrix Models,Theor. Math. Phys.162(2010) 1–33, [0906.3518]

work page arXiv 2010
[2]

De Concini and C

C. De Concini and C. Procesi,The Invariant Theory of Matrices, vol. 69 ofUniversity Lecture Series. American Mathematical Society, Providence, RI, 2017

work page 2017
[3]

The Hagedorn Transition, Deconfinement and N=4 SYM Theory

B. Sundborg,The hagedorn transition, deconfinement andN= 4SYM theory,Nucl. Phys. B 573(2000) 349–363, [hep-th/9908001]

work page Pith review arXiv 2000
[4]

A. M. Polyakov,Gauge fields and space-time,Int. J. Mod. Phys. A17S1(2002) 119–136, [hep-th/0110196]

work page arXiv 2002
[5]

The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theories

O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk,The Hagedorn - deconfinement phase transition in weakly coupled large N gauge theories,Adv. Theor. Math. Phys.8(2004) 603–696, [hep-th/0310285]

work page Pith review arXiv 2004
[6]

Makeenko and K

Y. Makeenko and K. Zarembo,Adjoint fermion matrix models,Nucl. Phys. B422(1994) 237–257, [hep-th/9309012]. – 43 –

work page arXiv 1994
[7]

G. W. Semenoff and R. J. Szabo,Fermionic matrix models,Int. J. Mod. Phys. A12(1997) 2135–2292, [hep-th/9605140]

work page arXiv 1997
[8]

Higher- order BRST and anti-BRST operators and cohomology for compact Lie algebras,

C. Chryssomalakos, J. A. de Azcarraga, A. J. Macfarlane and J. C. Perez Bueno,Higher order BRST and anti-BRST operators and cohomology for compact Lie algebras,J. Math. Phys.40(1999) 6009–6032, [hep-th/9810212]

work page arXiv 1999
[9]

L. D. Paniak and R. J. Szabo,Fermionic quantum gravity,Nucl. Phys. B593(2001) 671–725, [hep-th/0005128]

work page arXiv 2001
[10]

J. A. de Azcarraga and A. J. Macfarlane,Fermionic realizations of simple Lie algebras and their invariant fermionic operators,Nucl. Phys. B581(2000) 743–760, [hep-th/0003111]

work page arXiv 2000
[11]

Grassmann matrix quantum mechanics,

D. Anninos, F. Denef and R. Monten,Grassmann Matrix Quantum Mechanics,JHEP04 (2016) 138, [1512.03803]

work page arXiv 2016
[12]

Solvable quantum Grassmann matrices,

D. Anninos and G. A. Silva,Solvable Quantum Grassmann Matrices,J. Stat. Mech.1704 (2017) 043102, [1612.03795]

work page arXiv 2017
[13]

I. R. Klebanov, A. Milekhin, F. Popov and G. Tarnopolsky,Spectra of eigenstates in fermionic tensor quantum mechanics,Phys. Rev. D97(2018) 106023, [1802.10263]

work page arXiv 2018
[14]

Hagedorn temperature in largeNMajorana quantum mechanics,

G. Gaitan, I. R. Klebanov, K. Pakrouski, P. N. Pallegar and F. K. Popov,Hagedorn Temperature in LargeNMajorana Quantum Mechanics,Phys. Rev. D101(2020) 126002, [2002.02066]

work page arXiv 2020
[15]

Lie algebra fermions,

J. Troost,Lie Algebra Fermions,J. Phys. A53(2020) 425401, [2004.01055]

work page arXiv 2020
[16]

Fortuity with a single matrix,

Y. Chen,Fortuity with a single matrix,2511.00790

work page arXiv
[17]

Chevalley and S

C. Chevalley and S. Eilenberg,Cohomology Theory of Lie Groups and Lie Algebras,Trans. Am. Math. Soc.63(1948) 85–124

work page 1948
[18]

Koszul,Cohomology Theory of Lie Groups and Lie Algebras,Trans

J.-L. Koszul,Cohomology Theory of Lie Groups and Lie Algebras,Trans. Am. Math. Soc.63 (1948) 85–124

work page 1948
[19]

Kostant,Eigenvalues of a laplacian and commutative lie subalgebras,Topology3(1965) 147–159

B. Kostant,Eigenvalues of a laplacian and commutative lie subalgebras,Topology3(1965) 147–159

work page 1965
[20]

Berele,Trace identities andZ/2Z-graded invariants,Trans

A. Berele,Trace identities andZ/2Z-graded invariants,Trans. Amer. Math. Soc.309(1988) 581–589

work page 1988
[21]

Itoh,Twisted immanant and matrices with anticommuting entries, 2015

M. Itoh,Twisted immanant and matrices with anticommuting entries, 2015

work page 2015
[22]

Berele,Invariant theory for matrices over the grassmann algebra,Adv

A. Berele,Invariant theory for matrices over the grassmann algebra,Adv. Math.237(2013) 33–61

work page 2013
[23]

Y. P. Razmyslov,Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR Izv.8(1974) 727–760

work page 1974
[24]

Procesi,The invariant theory ofn×nmatrices,Adv

C. Procesi,The invariant theory ofn×nmatrices,Adv. Math.19(1976) 306–381

work page 1976
[25]

Counting chiral primaries in N = 1, d=4 superconformal field theories,

C. Romelsberger,Counting chiral primaries inN= 1,d= 4superconformal field theories, Nucl. Phys. B747(2006) 329–353, [hep-th/0510060]

work page arXiv 2006
[26]

An Index for 4 dimensional super conformal theories,

J. Kinney, J. M. Maldacena, S. Minwalla and S. Raju,An Index for 4 dimensional super conformal theories,Commun. Math. Phys.275(2007) 209–254, [hep-th/0510251]

work page arXiv 2007
[27]

Teranishi,The ring of invariants of matrices,Nagoya Math

Y. Teranishi,The ring of invariants of matrices,Nagoya Math. J.104(1986) 149–161

work page 1986
[28]

Pouliot,Molien function for duality,JHEP01(1999) 021, [hep-th/9812015]

P. Pouliot,Molien function for duality,JHEP01(1999) 021, [hep-th/9812015]. – 44 –

work page arXiv 1999
[29]

Benvenuti, B

S. Benvenuti, B. Feng, A. Hanany and Y.-H. He,Counting BPS operators in gauge theories: Quivers, syzygies and plethystics,JHEP11(2007) 050, [hep-th/0608050]

work page arXiv 2007
[30]

F. A. Dolan,Counting BPS operators inN= 4SYM,Nucl. Phys. B790(2008) 432–464, [0704.1038]

work page arXiv 2008
[31]

Murthy,Unitary matrix models, free fermion ensembles, and the giant graviton expansion, Pure Appl

S. Murthy,Unitary matrix models, free fermion ensembles, and the giant graviton expansion, Pure Appl. Math. Q.19(2023) 299–340, [2202.06897]

work page arXiv 2023
[32]

Rosselló,Finite-Nholography and the giant graviton expansion,Int

M. Rosselló,Finite-Nholography and the giant graviton expansion,Int. J. Mod. Phys. A41 (2026) 2648002

work page 2026
[33]

Murthy,The growth of the1/16-bps index in 4dN= 4SYM,Phys

S. Murthy,The growth of the1/16-bps index in 4dN= 4SYM,Phys. Rev. D105(2022) L021903, [2005.10843]

work page arXiv 2022
[34]

Chang and Y.-H

C.-M. Chang and Y.-H. Lin,Holographic covering and the fortuity of black holes, 2, 2024

work page 2024
[35]

Y. Chen, H. W. Lin and S. H. Shenker,BPS chaos,SciPost Phys.18(2025) 072, [2407.19387]

work page arXiv 2025
[36]

BPS spectra of $\operatorname{tr}[\Psi^p]$ matrix models for odd $p$

M. Tierz,BPS spectra oftr[Y p]matrix models for oddp,2604.27164

work page internal anchor Pith review Pith/arXiv arXiv
[37]

Ramgoolam,Permutations and the combinatorics of gauge invariants for general N,PoS CORFU2015(2016) 107, [1605.00843]

S. Ramgoolam,Permutations and the combinatorics of gauge invariants for general N,PoS CORFU2015(2016) 107, [1605.00843]

work page arXiv 2016
[38]

de Mello Koch and A

R. de Mello Koch and A. Jevicki,Hilbert space of finite N multi-matrix models,JHEP11 (2025) 145, [2508.11986]

work page arXiv 2025
[39]

Caputa and R

P. Caputa and R. de Mello Koch,Gauge invariants at arbitrary N and trace relations,JHEP 12(2025) 165, [2509.05834]

work page arXiv 2025
[40]

Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction

M. Berkooz, P. Narayan and J. Simon,Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction,JHEP08(2018) 192, [1806.04380]

work page Pith review arXiv 2018
[41]

Towards a full solution of the large N double-scaled SYK model

M. Berkooz, M. Isachenkov, V. Narovlansky and G. Torrents,Towards a full solution of the largeNdouble-scaled SYK model,JHEP03(2019) 079, [1811.02584]

work page Pith review arXiv 2019
[42]

Gaiotto and H

D. Gaiotto and H. Verlinde,SYK-Schur duality: double scaled SYK correlators fromN= 2 supersymmetric gauge theory,JHEP06(2025) 163, [2409.11551]

work page arXiv 2025
[43]

Procesi,Lie Groups: An Approach through Invariants and Representations

C. Procesi,Lie Groups: An Approach through Invariants and Representations. Universitext. Springer, New York, 2007. 10.1007/978-0-387-28929-8

work page doi:10.1007/978-0-387-28929-8 2007
[44]

Eleftheriou,A Holographic exploration of Superconformal Indices

G. Eleftheriou,A Holographic exploration of Superconformal Indices. PhD thesis, King’s Coll. London, 2025

work page 2025
[45]

Gadde, L

A. Gadde, L. Rastelli, S. S. Razamat and W. Yan,Gauge Theories and Macdonald Polynomials,Commun. Math. Phys.319(2013) 147–193, [1110.3740]

work page arXiv 2013
[46]

Arai and Y

R. Arai and Y. Imamura,FiniteNCorrections to the Superconformal Index of S-fold Theories,PTEP2019(2019) 083B04, [1904.09776]

work page arXiv 2019
[47]

Agarwal, S

P. Agarwal, S. Choi, J. Kim, S. Kim and J. Nahmgoong,AdS black holes and finiteN indices,Phys. Rev. D103(2021) 126006, [2005.11240]

work page arXiv 2021
[48]

The giant graviton expansion,

D. Gaiotto and J. H. Lee,The giant graviton expansion,JHEP08(2024) 025, [2109.02545]

work page arXiv 2024
[49]

J. H. Lee,Trace relations and open string vacua,JHEP02(2024) 224, [2312.00242]

work page arXiv 2024
[50]

J. H. Lee and D. Stanford,Bulk thimbles dual to trace relations,2412.20769. – 45 –

work page arXiv
[51]

de Mello Koch and A

R. de Mello Koch and A. Jevicki,Structure of loop space at finiteN,JHEP06(2025) 011, [2503.20097]

work page arXiv 2025
[52]

J. H. Lee and W. Li,AdS3 Quantum Gravity and FiniteNChiral Primaries,2511.00636

work page arXiv
[53]

A. S. Amitsur and J. Levitzki,Minimal identities for algebras,Proc. Amer. Math. Soc.1 (1950) 449–463

work page 1950
[54]

Procesi,On the theorem of amitsur–levitzki,Israel J

C. Procesi,On the theorem of amitsur–levitzki,Israel J. Math.207(2015) 151–154, [1308.2421]

work page arXiv 2015
[55]

I. G. Macdonald,Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. Oxford University Press, 2 ed., 1995

work page 1995
[56]

James and A

G. James and A. Kerber,The Representation Theory of the Symmetric Group, vol. 16 of Encyclopedia of Mathematics and its Applications. Addison-Wesley, Reading, Massachusetts, 1981

work page 1981
[57]

Zagier,The dilogarithm function, inFrontiers in Number Theory, Physics, and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization(P

D. Zagier,The dilogarithm function, inFrontiers in Number Theory, Physics, and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization(P. Cartier, P. Moussa, B. Julia and P. Vanhove, eds.), pp. 3–65. Springer Berlin Heidelberg, Berlin, Heidelberg, 2007. DOI

work page 2007
[58]

Budzik,Supergroup Invariants and the Brane/Negative Brane Expansion,2509.20451

K. Budzik,Supergroup Invariants and the Brane/Negative Brane Expansion,2509.20451

work page arXiv
[59]

The giant graviton expansion in AdS5×S5,

G. Eleftheriou, S. Murthy and M. Rosselló,The giant graviton expansion inAdS5 ×S 5, SciPost Phys.17(2024) 098, [2312.14921]

work page arXiv 2024
[60]

Eleftheriou, S

G. Eleftheriou, S. Murthy and M. Rosselló,Localization and wall-crossing of giant graviton expansions in AdS5,JHEP07(2025) 126, [2501.13910]

work page arXiv 2025
[61]

Zeilberger and D

D. Zeilberger and D. M. Bressoud,A proof of andrews’q-dyson conjecture,Discrete Math.54 (1985) 201–224

work page 1985
[62]

D. M. Bressoud and I. P. Goulden,Constant term identities extending theq-dyson theorem, Trans. Amer. Math. Soc.291(1985) 203–228

work page 1985
[63]

Zagier,Applications of the Representation Theory of Finite Groups, vol

D. Zagier,Applications of the Representation Theory of Finite Groups, vol. 141 of Encyclopaedia of Mathematical Sciences, pp. 399–427. Springer-Verlag, Berlin, Heidelberg, 2004

work page 2004
[64]

Regev,Sign trace identities,Linear and Multilinear Algebra21(1987) 1–28

A. Regev,Sign trace identities,Linear and Multilinear Algebra21(1987) 1–28

work page 1987
[65]

Berele,Invariant theory and trace identities associated with Lie color algebras,Journal of Algebra310(2007) 194–206

A. Berele,Invariant theory and trace identities associated with Lie color algebras,Journal of Algebra310(2007) 194–206

work page 2007
[66]

Zhang,The first and second fundamental theorems of invariant theory for the quantum general linear supergroup,Journal of Pure and Applied Algebra224(2020) 106411

Y. Zhang,The first and second fundamental theorems of invariant theory for the quantum general linear supergroup,Journal of Pure and Applied Algebra224(2020) 106411

work page 2020
[67]

Razzaq, R

J. Razzaq, R. Fioresi and M. A. Lledó,On fundamental theorems of invariant theory for the special linear supergroup, 2025

work page 2025
[68]

Nakamoto,The structure of the invariant ring of two matrices of degree 3,Journal of Pure and Applied Algebra166(2002) 125–148

K. Nakamoto,The structure of the invariant ring of two matrices of degree 3,Journal of Pure and Applied Algebra166(2002) 125–148

work page 2002
[69]

Drensky and L

V. Drensky and L. Sadikova,Generators of invariants of two4×4matrices,C. R. Acad. Bulgare Sci.59(2006) 477–484, [math/0503146]

work page arXiv 2006
[70]

Aslaksen, V

H. Aslaksen, V. Drensky and L. Sadikova,Defining relations of invariants of two3×3 matrices,J. Algebra298(2006) 41–57, [math/0405389]. – 46 –

work page arXiv 2006
[71]

Benanti and V

F. Benanti and V. Drensky,Defining relations of minimal degree of the trace algebra of3×3 matrices,J. Algebra320(2008) 756–782, [math/0701609]

work page arXiv 2008
[72]

Drensky and R

V. Drensky and R. La Scala,Defining relations of low degree of invariants of two4×4 matrices,Int. J. Algebra Comput.19(2009) 107–127, [0708.3583]

work page arXiv 2009
[73]

A. A. Lopatin,On the nilpotency degree of the algebra with identityxn = 0,J. Algebra371 (2012) 350–366, [1106.0950]

work page arXiv 2012
[74]

Hoge,A presentation of the trace algebra of three3×3matrices,Journal of Algebra358 (2012) 257–268

T. Hoge,A presentation of the trace algebra of three3×3matrices,Journal of Algebra358 (2012) 257–268

work page 2012
[75]

Derksen and V

H. Derksen and V. Makam,Polynomial degree bounds for matrix semi-invariants,Adv. Math. 310(2017) 44–63, [1512.03393]

work page arXiv 2017
[76]

Berele,GL n-invariant functions onMn(G),Turkish J

A. Berele,GL n-invariant functions onMn(G),Turkish J. Math.46(2022) 1667–1673

work page 2022
[77]

García-Martínez, Z

X. García-Martínez, Z. Normatov and R. Turdibaev,The ring of invariants of pairs of3×3 matrices,Journal of Algebra603(2022) 201–212

work page 2022
[78]

Eshmatov, X

F. Eshmatov, X. García-Martínez and R. Turdibaev,Noncommutative poisson structure and invariants of matrices,Advances in Mathematics469(2025) 110212

work page 2025
[79]

Brešar, C

M. Brešar, C. Procesi and Š. Špenko,Quasi-identities on matrices and the cayley–hamilton polynomial,Adv. Math.280(2015) 439–471, [1212.4597]

work page arXiv 2015
[80]

G. E. Andrews and K. Eriksson,Integer Partitions. Cambridge University Press, Cambridge, 2004. – 47 –

work page 2004