arxiv: 2604.25462 · v1 · submitted 2026-04-28 · 🧮 math.RT · math.QA

Recognition: unknown

Representations of Super Yangians with Gelfand-Tsetlin bases

Jian Zhang, Vyacheslav Futorny, Zheng Li

Pith reviewed 2026-05-07 14:10 UTC · model grok-4.3

classification 🧮 math.RT math.QA

keywords super Yangiantame moduleGelfand-Tsetlin basisevaluation homomorphismcovariant tensor modulehighest weight vectorLie superalgebra gl(m|n)

0 comments

The pith

Super Yangian modules from tensor products of covariant tensors are tame precisely when a highest-weight condition holds.

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

The paper uses evaluation homomorphisms to turn covariant tensor modules of the Lie superalgebra gl_{m|n} into modules over the super Yangian Y_{m|n}. It then examines the submodule generated inside a tensor product by products of highest weight vectors and takes the simple quotient. When the odd parameter n is zero, the same construction recovers every finite-dimensional simple module of the ordinary Yangian Y(gl_m). The main result supplies a necessary and sufficient condition on the highest weights that makes these quotients tame. A reader cares because the condition extends the known Nazarov-Tarasov criterion and thereby gives an explicit handle on a large family of representations for these infinite-dimensional superalgebras.

Core claim

The evaluation homomorphisms from the super Yangian Y_{m|n} to the universal enveloping algebra U(gl_{m|n}) allow one to regard the covariant tensor modules of gl_{m|n} as Y_{m|n}-modules. The paper studies simple quotients of the submodules generated by a tensor product of highest weight vectors inside tensor products of these evaluation modules. In the case n=0 this recovers all finite-dimensional simple modules of Y(gl_m). The central result is a necessary and sufficient condition for such modules to be tame.

What carries the argument

The evaluation homomorphism from Y_{m|n} to U(gl_{m|n}) together with the submodule generated by products of highest weight vectors inside tensor products of covariant evaluation modules, whose simple quotients are tame precisely under the stated highest-weight condition.

If this is right

When n=0 the construction produces every finite-dimensional simple module of the ordinary Yangian Y(gl_m).
The tameness criterion generalizes the Nazarov-Tarasov condition from the non-super case to Y_{m|n}.
Simple quotients that meet the condition admit an explicit description in terms of their highest weights.
Modules whose highest weights violate the condition are not tame.

Where Pith is reading between the lines

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

The same highest-weight criterion may extend to other families of super Yangians or to quantized enveloping algebras.
When the condition holds, the Gelfand-Tsetlin bases furnished by tameness could be used to compute characters or branching rules for these modules.
Checking the condition on low-rank cases (small m and n) would produce explicit lists of tame modules that can be compared with known classifications.

Load-bearing premise

The evaluation homomorphisms exist and convert the covariant tensor modules into well-defined Y_{m|n}-modules whose tensor products contain the described submodules and simple quotients.

What would settle it

A concrete highest-weight vector in a small tensor product (for example m=2, n=1) for which the stated condition holds but the corresponding simple quotient fails to be tame, or vice versa.

read the original abstract

The evaluation homomorphisms from the super Yangian $\Ymn$ to the universal enveloping algebra $\U(\gl_{m|n})$ allows one to regard the covariant tensor module of $\gl_{m|n}$ as $\Ymn$ modules. We study simple quotients of the submodules generated by a tensor product of highest weight vectors inside the tensor products of covariant evaluation modules. In the case $n=0$, this recover all finite-dimensional simple modules of $\Y(\mathfrak{gl}_m)$. We give a necessary and sufficient condition for such modules to be tame, which generalizes the earlier work of Nazarov and Tarasov for $\Y(\mathfrak{gl}_m)$ to the super case.

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

This paper gives a necessary and sufficient tameness condition for certain simple quotients over super Yangians Y_{m|n}, extending the Nazarov-Tarasov result to the super case.

read the letter

The main point is that the authors establish a necessary and sufficient condition for tameness of simple quotients of submodules generated by tensor products of highest weight vectors inside covariant evaluation modules for the super Yangian. This directly generalizes the earlier classification for ordinary Yangians and recovers it when n=0. They construct the corresponding Gelfand-Tsetlin bases via the super version of the usual rules. The argument rests on the standard evaluation homomorphisms from Y_{m|n} to U(gl_{m|n}).

Referee Report

0 major / 2 minor

Summary. The manuscript uses evaluation homomorphisms from the super Yangian Y_{m|n} to U(gl_{m|n}) to equip covariant tensor modules of gl_{m|n} with Y_{m|n}-module structures. It studies the simple quotients of submodules generated by tensor products of highest weight vectors inside tensor products of these evaluation modules. When n=0 the construction recovers all finite-dimensional simple modules of Y(gl_m). The central result is a necessary and sufficient condition for tameness of the resulting modules, together with an explicit construction of Gelfand-Tsetlin bases via the super-analog of the usual combinatorial rules; this generalizes the Nazarov-Tarasov classification to the super setting.

Significance. If the proofs are complete, the work supplies a concrete generalization of a foundational result on tame Yangian modules to the super case. The recovery of the classical n=0 case supplies an independent consistency check, and the Gelfand-Tsetlin basis construction furnishes a combinatorial tool that should permit explicit character computations and branching rules. The approach relies on standard evaluation homomorphisms and prior results rather than ad-hoc inventions, which strengthens its potential utility for further study of super Yangian representations and their applications in supersymmetric integrable systems.

minor comments (2)

Abstract: the notation Ymn (rendered as Y_{m|n}) is used without an explicit definition; a parenthetical clarification would improve readability for readers outside the immediate subfield.
The manuscript should include a short dedicated paragraph or subsection that states the precise necessary-and-sufficient tameness condition in combinatorial terms (e.g., in terms of the super Young diagram or the highest-weight parameters) so that the generalization of Nazarov-Tarasov is immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report, so we have no points requiring point-by-point rebuttal or clarification. We will make any minor editorial improvements in the revised version as needed.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs representations of the super Yangian Y_{m|n} via standard evaluation homomorphisms to U(gl_{m|n}), realizes covariant tensor modules as Y_{m|n}-modules, and identifies simple quotients of submodules generated by highest weight vectors in tensor products. It then states a necessary and sufficient tameness condition that directly generalizes the Nazarov-Tarasov criterion (recovered when n=0). No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or load-bearing self-citation; the Gelfand-Tsetlin bases follow combinatorial rules analogous to the classical case, and the tameness criterion is formulated in terms of external algebraic data rather than internal fits. The n=0 recovery serves as an independent consistency check against known results, confirming the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definitions and properties of super Yangians, evaluation homomorphisms, and covariant tensor modules of gl(m|n); no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

domain assumption Evaluation homomorphisms from the super Yangian Y_{m|n} to U(gl_{m|n}) exist and allow covariant tensor modules to be viewed as Y_{m|n}-modules.
Invoked to regard the covariant tensor modules of gl_{m|n} as Ymn modules.
domain assumption Tensor products of these modules contain submodules generated by products of highest weight vectors whose simple quotients are well-defined.
Central to the construction studied in the abstract.

pith-pipeline@v0.9.0 · 5418 in / 1525 out tokens · 95228 ms · 2026-05-07T14:10:06.647160+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references

[1]

Berele and A

A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. in Math. 64 (1987), no. 2, 118–175. 2

1987
[2]

Cherednik, A new interpretation of Gel’fand-Tzetlin bases, Duke Math. J. 54 (1987), no. 2, 563–577. 1

1987
[3]

Futorny, Vyacheslav, V era Serganova, and Jian Zhang, Gelfand-Tsetlin modules for gl(m|n) , Mathematical Research Letters 28.5 (2022): 1379-1418. 10

2022
[4]

Gow, On the Yangian Y (glm|n) and its quantum Berezinian , Czechoslovak J

L. Gow, On the Yangian Y (glm|n) and its quantum Berezinian , Czechoslovak J. Phys. 55 (2005), no. 11, 1415–1420. 5

2005
[5]

Gow, Gauss decomposition of the Yangian Y (glm|n), Comm

L. Gow, Gauss decomposition of the Yangian Y (glm|n), Comm. Math. Phys. 276 (2007), no. 3, 799–825. 5, 7, 8

2007
[6]

Lu, Gelfand-Tsetlin bases of representations for super Yangian and quantum afﬁne superalgebra , Lett

K. Lu, Gelfand-Tsetlin bases of representations for super Yangian and quantum afﬁne superalgebra , Lett. Math. Phys. 111 (2021), no. 6, Paper No. 145, 30 pp. 2

2021
[7]

Lu and E

K. Lu and E. E. Mukhin, Jacobi-Trudi identity and Drinfeld functor for super Yangian , Int. Math. Res. Not. IMRN 2021, no. 21, 16751–16810. 2, 20

2021
[8]

A. I. Molev, Gelfand-Tsetlin basis for representations of Y angians, Lett. Math. Phys. 30 (1994), no. 1, 53–60. 1, 12

1994
[9]

Molev, Yangians for classical Lie algebras

A. Molev, Yangians for classical Lie algebras. Mathematical Surveys and Monographs 143. Amer. Math. Soc., Providence, RI, 2007. 1, 8

2007
[10]

Molev, Combinatorial bases for covariant representations of the Lie superalgebra glm|n, Bull

A. Molev, Combinatorial bases for covariant representations of the Lie superalgebra glm|n, Bull. Inst. Math. Acad. Sin. (N.S.) 6 (2011), no. 4, 415–462. 2, 10

2011
[11]

A. I. Molev, M. Nazarov and G. I. Ol’shanski ˘i, Yangians and classical Lie algebras, Russian Math. Surveys 51 (1996), no. 2, 205–282; translated from Uspekhi Mat. Nauk 51 (1996), no. 2(308), 27–104. 1

1996
[12]

Nazarov, Quantum Berezinian and the classical Capelli identity , Lett

M. Nazarov, Quantum Berezinian and the classical Capelli identity , Lett. Math. Phys. 21 (1991), no.2, 123–

1991
[13]

Nazarov, Yangian of the general linear Lie superalgebra, SIGMA Symmetry Integrability Geom

M. Nazarov, Yangian of the general linear Lie superalgebra, SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No. 112, 24 pages. 6, 7, 8

2020
[14]

Nazarov and V

M. Nazarov and V . O. Tarasov, Yangians and Gel’fand-Zetlin bases , Publ. Res. Inst. Math. Sci. 30 (1994), no. 3, 459–478. 1, 12

1994
[15]

Nazarov and V

M. Nazarov and V . Tarasov, Representations of Yangians with Gelfand-Zetlin bases , J. Reine Angew. Math. 496 (1998), 181–212. 1, 2, 3, 12, 20, 21, 25, 29

1998
[16]

T. D. Palev,Irreducible ﬁnite-dimensional representations of the Lie superalgebra gl(n/1) in a Gel’fand-Zetlin basis, J. Math. Phys. 30 (1989), no. 7, 1433–1442. 2, 9

1989
[17]

T. D. Palev,Essentially generic representations of the Lie superalgebras gl(n|m) in the Gel’fand-Tsetlin basis, Funct. Anal. Appl. 23 (1989), no. 2, 141–142; translated from Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 69–70. 2, 9, 10

1989
[18]

A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras Gl(n, m ) and Q(n), Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422–430. 2 30 VY ACHESLA V FUTORNY, ZHENG LI, AND JIAN ZHANG

1984
[19]

Stoilova and J

N. Stoilova and J. V an der Jeugt, Gel’fand-Zetlin basis and Clebsch-Gordan coefﬁcients for covariant repre- sentations of the Lie superalgebra gl(m|n), J. Math. Phys. 51 (2010), no. 9, 093523, 15. 2, 9, 10

2010
[20]

Zhang, Representations of super Yangian, J

R. Zhang, Representations of super Yangian, J. Math. Phys. 36 (1995), no. 7, 3854–3865. 1

1995
[21]

Zhang, The gl(M |N ) super Yangian and its ﬁnite-dimensional representations, Lett

R. Zhang, The gl(M |N ) super Yangian and its ﬁnite-dimensional representations, Lett. Math. Phys. 37 (1996), no. 4, 419–434. 1, 20 SHENZHEN INTERNATIONAL CENTER FOR MATHEMATICS , S OUTHERN UNIVERSITY OF SCIENCE AND TECH - NOLOGY , S HENZHEN , C HINA Email address: vfutorny@gmail.com SCHOOL OF ARTIFICIAL INTELLIGENCE , J IANGHAN UNIVERSITY , W UHAN , H UB...

1996