The six-vertex model at roots of unity and some highest weight representations of the sl(2) loop algebra

Tetsuo Deguchi

REVIEW

The six-vertex model at roots of unity and some highest weight representations of the sl(2) loop algebra

Tetsuo Deguchi

Statistical Mechanics · Submitted 6 Mar 2006 · 20 references

Pith's verdict: An honest but thin paper: the advertised general criterion is stated without proof, and the elementary proof of the known theorem rests on an unverified q->1 limit.

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arXiv IDcond-mat/0603112 (version 1)

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CitationTetsuo Deguchi. "The six-vertex model at roots of unity and some highest weight representations of the sl(2) loop algebra." arXiv:cond-mat/0603112 (2006). https://pith.science/paper/EET64I4M

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Hi — quick take on Deguchi's loop algebra paper. The headline: it does not prove what it advertises. Theorem 2, the general irreducibility criterion, is stated in the abstract but its proof is deferred to a future report. The only fully proved theorem, Theorem 1, is already known (Chari–Pressley [5]), and the author cites it. So the genuinely new content is the elementary proof and the explicit 8-dimensional reducible indecomposable example.

The elementary proof itself has a clear soft spot. Lemma 3's reduction relation (18) is derived from (19), and the paper says (19) was shown for U_q(L(sl2)) in [3], then applies it directly to the classical loop algebra. That is not immediate: the q-deformed commutators and divided powers need a q -> 1 limit that is not supplied. If (19) does not hold classically, the annihilation relation (20) and the whole basis construction collapse. The theorem may still be true — Chari–Pressley supports it — but this paper's proof is incomplete as written. I would not call it fatal, since the result has independent support, but it is a real gap in a central step.

The example in Proposition 2 is the part I found most useful. The 8-dimensional reducible indecomposable module with parameters (a1,a1,a2) is concrete and could matter for spectral degeneracies of the XXZ chain at roots of unity. But several claims — rho1^3 Omega != 0, rho2^2 Omega = 0, rho1 w1 Omega = 0 — are asserted without derivation. They look plausible, but the paper does not show the work.

On citations: the paper cites its own previous work and Chari–Pressley appropriately, and self-citation is not a problem here. The abstract does overstate the contribution by presenting Theorem 2 as a proved result when the proof is explicitly absent.

Bottom line: this is for someone working on loop algebra symmetry of integrable models who wants an explicit example. It is not a major new result, but it is honest about the deferred proof, and the example is worth having. I would send it to a referee if the journal will accept a modest contribution with a request to fill gaps: the author should prove the classical analogue of (19) or state Lemma 3 as an assumption, and either prove Theorem 2 or move it to a clearly marked conjecture. As it stands I would not rely on it, but I would engage with a revised version.

Referee report

Significance

If completed, the paper offers a self-contained computational route to a result already known from Chari-Pressley [5], and the rho-basis construction could be a useful tool for computing dimensions of highest weight representations that appear as Bethe-state degeneracies. The paper also identifies a concrete criterion (x^-_s(a)Omega = 0) for irreducibility and illustrates a reducible indecomposable representation. The physical motivation is clear and the topic is relevant to the journal's scope. However, the proof as written has a genuine gap: Lemma 3 imports a key relation from the quantum affine setting without proving its classical analogue, and Theorem 2 and the worked example rely on asserted statements whose derivations are not supplied. Because the central theorem has independent support in the literature, I do not doubt the result, but the manuscript's elementary proof is currently incomplete.

Major objections (3)

[Sec. 3.1, Lemma 3 and Eqs. (18)-(20)] The reduction relation (18) is proved by invoking relation (19), which is said to have been shown for U_q(L(sl(2))) in [3]. The paper applies it directly to the classical loop algebra L(sl2) without proving the classical analogue. This is not a routine substitution: (19) involves divided powers and structures that are not immediate from the defining relations (5). Relations (18) and (20) are load-bearing: (20), x^-_r(\hat A_phi)Omega = 0, is used in Lemma 4 and in the basis argument leading to Lemma 6 and Proposition 1, and Proposition 1 is the foundation of the proof of Theorem 1. Please provide a direct proof of (18) in L(sl2) or a reference that contains the classical version.
[Sec. 3.3, Theorem 2] Theorem 2 is stated as a theorem, but its proof is explicitly deferred to the next report [9]. Since the abstract advertises "a general criterion for a highest weight representation to be irreducible", and the following paragraph applies this criterion to Bethe states, presenting it as a proved theorem is not appropriate in the current manuscript. If the proof cannot be included, the statement should be labeled a conjecture, or the paper should be revised to remove the claim of a proved criterion.
[Sec. 3.3, example and Proposition 2, Eqs. (30)-(32)] Several assertions that are essential for the example are made without derivation: x^+_n w1 Omega = 0 for all n (Eq. (30)); rho_1^3 Omega != 0, rho_1^4 Omega = 0, rho_2 Omega != 0, rho_2^2 Omega = 0, w_1^2 Omega = 0 (Eq. (31)); and rho_1 w_1 Omega = 0 together with rho_2 w_1 Omega != 0 when w_1 Omega != 0 (Eq. (32)). These relations determine the sector bases and hence the dimension count in Proposition 2. They are not side remarks. Please supply derivations or explicit references for each of these statements, or prove Proposition 2 by another argument.

Minor objections (5)

[Abstract and Sec. 1] The word "irreducble" in the abstract should be "irreducible".
[Sec. 2, Eq. (6)] In the displayed definition of h_m(A), the symbol "h^\pm_{m-k}" appears; since no sign index is intended in h_m(A), this should be "h_{m-k}".
[Sec. 3.3, Conjecture 1 and Proposition 2] The expression "dimensions 2r" in Conjecture 1 and "23 dimensions" near Proposition 2 should presumably read "2^r" and "2^3", respectively; please correct these.
[Secs. 1 and 3.1] The barred notation \bar{h}_0 and \bar{d}^\pm_0 appears without definition; please define these symbols in the text.
[References] Reference [8] is cited as "cond-mat/0503564, 2nd version", which is not a complete formal reference; a journal or arXiv identifier with full bibliographic data should be provided.

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The paper

The paper claims

Every highest weight representation of the sl2 loop algebra with distinct evaluation parameters is irreducible, making each Bethe-state degeneracy in the root-of-unity six-vertex model exactly computable.

The reading

This paper proves that every highest weight representation of the sl(2) loop algebra with distinct evaluation parameters is irreducible, and it does so by an elementary route that constructs an explicit basis for each sector of the Cartan generator $h_0$. The setting is the six-vertex model, equivalently the XXZ spin chain, at roots of unity, where the transfer matrix commutes with the sl(2) loop algebra and regular Bethe ansatz eigenstates are highest weight vectors. If the theorem is right, the number of linearly independent degenerate states accompanying each such eigenstate is exactly the dimension of an irreducible highest weight module, computed from the Drinfeld polynomial. The paper also states a general criterion, $V_\Omega$ irreducible if and only if $x^-_s(a)\Omega=0$, and illustrates the degenerate case with evaluation parameters $a_1,a_1,a_2$, where the module is reducible indecomposable of dimension $2^3=8$ when a certain vector $w_1\Omega$ is nonzero, and irreducible of dimension 6 when it vanishes.

What carries the argument

The central identity is the reduction relation (18), $x^-_{\ell+r+1}\Omega=\sum_{j=1}^{r}(-1)^{r-j}\lambda_{r+1-j}x^-_{\ell+j}\Omega$, derived from the algebraic relation (19) imported from the quantum affine algebra setting. It implies $x^-_r(\hat A_\varphi)\Omega=0$ when evaluation parameters have multiplicities, and in the distinct case it reduces every lowering monomial to an ordered product of $\rho_j(a)=x^-_{s-1}(A_j)$. Linear independence of these products is shown by acting with powers of $x^+$ and appealing to the Vandermonde determinant, and this explicit basis is what makes the irreducibility proof elementary.

Extended reading notes

Core claim

Let $\Omega$ be a highest weight vector of the sl2 loop algebra, with Drinfeld polynomial $P_\Omega(u)=\prod_{k=1}^{s}(1-a_k u)^{m_k}$; the distinct numbers $a_j$ are the evaluation parameters and $r=m_1+\cdots+m_s$. The paper's main theorem states that when the $a_j$ are distinct, the representation $V_\Omega$ generated by $\Omega$ is irreducible: every nonzero vector can be sent to $\Omega$ by some product of raising generators, and the sector of $h_0$-weight $r-2n$ has basis $\rho_{j_1}(a)\cdots\rho_{j_n}(a)\Omega$ with $1\le j_1\le\cdots\le j_n\le s$, where $\rho_j(a)=x^-_{s-1}(A_j)$ and $A_j$ is the list of evaluation parameters with $a_j$ removed. This is an alternative, constructive proof of a statement previously obtained through Weyl-module theory. The paper further states that irreducibility holds if and only if $x^-_s(a)\Omega=0$, with the proof postponed; and it constructs the $r=3$ example with repeated parameter $a_1$ to show that when the annihilation fails, the representation can be reducible and indecomposable of dimension 8.

Load-bearing premise

The load-bearing premise is that the commutation relation (19), proved for the quantum affine algebra $U_q(L(\mathrm{sl}_2))$, also holds in the classical $\mathrm{sl}_2$ loop algebra; the paper cites the quantum result without proving the classical analogue, and the reduction relation, the annihilation $x^-_r(\hat A_\varphi)\Omega=0$, and the entire basis construction depend on it.

Editorial extensions

If this is right

In the sector $S^Z\equiv 0\ (\mathrm{mod}\ N)$ of the root-of-unity six-vertex model, every regular Bethe eigenvector with distinct evaluation parameters generates an irreducible sl2 loop algebra module, so its spectral degeneracy is the known dimension of that irreducible highest weight module.
Whenever $x^-_s(a)\Omega=0$ holds for a Bethe eigenvector, even with repeated evaluation parameters, the stated criterion makes the module irreducible and its dimension follows from the Drinfeld polynomial.
Repeated evaluation parameters can instead produce reducible indecomposable modules: for $r=3$ with parameters $\{a_1,a_1,a_2\}$, the paper exhibits a module of dimension 8 when $w_1\Omega\neq 0$ and dimension 6 when $w_1\Omega=0$.
The explicit basis of $\rho$-products gives a direct count of the number of linearly independent states obtained by applying the lowering operators $S^-(N)$ and $T^-(N)$ to a highest weight Bethe state.
If the proposed conjecture holds, a highest weight module whose lowering operators $x^-_{r-1}(\hat A_j)$ fail to annihilate $\Omega$ for every $j$ is reducible indecomposable of dimension $2^r$, giving a power-of-two degeneracy pattern.

Reading between the lines

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

A direct verification of relation (19) in the classical, undeformed loop algebra $L(\mathrm{sl}_2)$ would test the paper's main proof step; the quantum proof cited for it may or may not survive the classical limit, and the relation could also be checked numerically against explicit matrices for small $r$.
The criterion "irreducible iff $x^-_s(a)\Omega=0$" suggests a practical state-by-state algorithm for the six-vertex model: evaluate this annihilation condition on regular Bethe solutions, and the outcome classifies each degeneracy, with failure of the condition feeding the $2^r$ prediction of Conjecture 1.
The Vandermonde-basis technique is not specific to sl2 and could plausibly extend to Drinfeld realizations of Yangians or quantum affine algebras, or to higher-rank loop algebras, as a general recipe for explicit bases of evaluation modules.
The $2^r$ dimension pattern in the degenerate example hints that repeated evaluation parameters create nilpotent lowering actions whose combinatorial footprint is binary; if true, degeneracies in the root-of-unity six-vertex model would always be powers of two.

Original abstract

We discuss irreducible highest weight representations of the sl(2) loop algebra and reducible indecomposable ones in association with the sl(2) loop algebra symmetry of the six-vertex model at roots of unity. We formulate an elementary proof that every highest weight representation with distinct evaluation parameters is irreducible. We present a general criteria for a highest weight representation to be irreducble. We also give an example of a reducible indecomposable highest weight representation and discuss its dimensionality.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We discuss irreducible highest weight representations of the sl(2) loop algebra and reducible indecomposable ones in association with the sl(2) loop algebra symmetry of the six-vertex model at roots of unity. We formulate an elementary proof that every highest weight representation with distinct evaluation parameters is irreducible. We present a general criteria for a highest weight representation to be irreducble. We also give an example of a reducible indecomposable highest weight representation and discuss its dimensionality.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We discuss irreducible highest weight representations of the sl(2) loop algebra and reducible indecomposable ones in association with the sl(2) loop algebra symmetry of the six-vertex model at roots of unity.	contribution 0.75	0.77	0.18	asserted 0.84
s1	We formulate an elementary proof that every highest weight representation with distinct evaluation parameters is irreducible.	result 0.51	0.95	0.31	asserted 1.00
s2	We present a general criteria for a highest weight representation to be irreducble.	contribution 0.68	0.93	0.21	asserted 1.00
s3	We also give an example of a reducible indecomposable highest weight representation and discuss its dimensionality.	contribution 0.69	0.93	0.09	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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References

Works cited (20)

[5]

V. Chari and A. Pressley, Weyl modules for classical and q uantum aﬃne algebras, Representation Theory 5 (2001) 191–223

View on Pith 2001
[3]

V. Chari and A. Pressley, Quantum Aﬃne Algebras, Commun. Math. Phys. 142 (1991) 261–283

View on Pith 1991
[9]

T. Deguchi, Dimensions of highest weight representatio ns of the sl2 loop algebra, in preparation

View on Pith
[1]

D. Arnaudon and V. Rittenberg, Quantum chains with Uq(SL(2)) symmetry and unrestricted representations, Phys. Lett. B 306 (1993) 86–90

View on Pith 1993
[2]

R. J. Baxter, Completeness of the Bethe ansatz for the six and eight vertex models, J. Stat. Phys. 108 (2002) 1–48

View on Pith 2002
[4]

V. Chari and A. Pressley, Quantum Aﬃne Algebras at Roots o f Unity, Representa- tion Theory 1 (1997) 280–328

View on Pith 1997
[6]

E. Date, M. Jimbo, K. Miki and T. Miwa, Generalized chiral Potts models and minimal cyclic representations of Uq( ˆgl(n, C), Commun. Math. Phys. 137 (1991) 133–147

View on Pith 1991
[7]

C. De Concini and V. Kac, Representations of quantum grou ps at roots of 1, in Oper- ator Algebras, Unitary Representations, Enveloping Algeb ras and Invariant Theory , A. Connes, M. Duﬂo, A. Joseph and R. Rentschler (eds), (Birkh ¨ auser, Boston, 1990) pp. 471–506. 9

View on Pith 1990
[8]

T. Deguchi, XXZ Bethe states as highest weight vectors of the sl2 loop algebra at roots of unity . (cond-mat/0503564, 2nd version)

View on Pith Pith review arXiv
[10]

T. Deguchi, K. Fabricius and B. M. McCoy, The sl2 Loop Algebra Symmetry of the Six-Vertex Model at Roots of Unity, J. Stat. Phys. 102 (2001) 701–736

View on Pith 2001
[11]

V.G. Drinfeld, A new realization of Yangians and quanti zed aﬃne algebras, Soviet Math. Doklady 36 (1988) 212–216

View on Pith 1988
[12]

K. Fabricius and B. M. McCoy, Bethe’s Equation Is Incomp lete for the XXZ Model at Roots of Unity, J. Stat. Phys. 103(2001) 647–678

View on Pith 2001
[13]

K. Fabricius and B. M. McCoy, Completing Bethe’s Equati ons at Roots of Unity, J. Stat. Phys. 104(2001) 573–587

View on Pith 2001
[14]

K. Fabricius and B. M. McCoy, Evaluation Parameters and Bethe Roots for the Six-Vertex Model at Roots of Unity, in Progress in Mathemati cal Physics Vol. 23 (MathPhys Odyssey 2001 ), edited by M. Kashiwara and T. Miwa, (Birkh¨ auser, Boston, 2002) 119–144

View on Pith 2001
[15]

M. Jimbo, private communication, July 2004

View on Pith 2004
[16]

G. Lusztig, Modular representations and quantum group s, Contemp. Math. 82 (1989) 59–77

View on Pith 1989
[17]

G. Lusztig, Introduction to Quantum Groups (Birkh¨ auser, Boston, 1993)

View on Pith 1993
[18]

V. Pasquier and H. Saleur, Common structures between ﬁn ite systems and confor- mal ﬁeld theories through quantum groups, Nucl. Phys. B 330 (1990) 523–556

View on Pith 1990
[19]

P. Roche and D. Arnaudon, Irreducible representations of the quantum analogue of SU(2), Lett. Math. Phys. 17 (1989) 295–300

View on Pith 1989
[20]

V. O. Tarasov, Cyclic Monodromy matrices for the R-matrix of the six-vertex model and the chiral Potts model with ﬁxed spin boundary condition s, Int. J. Mod. Phys. A 7 Suppl. 1B (1992) 963–975. 10

View on Pith 1992

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [5]

V. Chari and A. Pressley, Weyl modules for classical and q uantum aﬃne algebras, Representation Theory 5 (2001) 191–223

View on Pith 2001

[2] [3]

V. Chari and A. Pressley, Quantum Aﬃne Algebras, Commun. Math. Phys. 142 (1991) 261–283

View on Pith 1991

[3] [9]

T. Deguchi, Dimensions of highest weight representatio ns of the sl2 loop algebra, in preparation

View on Pith

[4] [1]

D. Arnaudon and V. Rittenberg, Quantum chains with Uq(SL(2)) symmetry and unrestricted representations, Phys. Lett. B 306 (1993) 86–90

View on Pith 1993

[5] [2]

R. J. Baxter, Completeness of the Bethe ansatz for the six and eight vertex models, J. Stat. Phys. 108 (2002) 1–48

View on Pith 2002

[6] [4]

V. Chari and A. Pressley, Quantum Aﬃne Algebras at Roots o f Unity, Representa- tion Theory 1 (1997) 280–328

View on Pith 1997

[7] [6]

E. Date, M. Jimbo, K. Miki and T. Miwa, Generalized chiral Potts models and minimal cyclic representations of Uq( ˆgl(n, C), Commun. Math. Phys. 137 (1991) 133–147

View on Pith 1991

[8] [7]

C. De Concini and V. Kac, Representations of quantum grou ps at roots of 1, in Oper- ator Algebras, Unitary Representations, Enveloping Algeb ras and Invariant Theory , A. Connes, M. Duﬂo, A. Joseph and R. Rentschler (eds), (Birkh ¨ auser, Boston, 1990) pp. 471–506. 9

View on Pith 1990

[9] [8]

T. Deguchi, XXZ Bethe states as highest weight vectors of the sl2 loop algebra at roots of unity . (cond-mat/0503564, 2nd version)

View on Pith Pith review arXiv

[10] [10]

T. Deguchi, K. Fabricius and B. M. McCoy, The sl2 Loop Algebra Symmetry of the Six-Vertex Model at Roots of Unity, J. Stat. Phys. 102 (2001) 701–736

View on Pith 2001

[11] [11]

V.G. Drinfeld, A new realization of Yangians and quanti zed aﬃne algebras, Soviet Math. Doklady 36 (1988) 212–216

View on Pith 1988

[12] [12]

K. Fabricius and B. M. McCoy, Bethe’s Equation Is Incomp lete for the XXZ Model at Roots of Unity, J. Stat. Phys. 103(2001) 647–678

View on Pith 2001

[13] [13]

K. Fabricius and B. M. McCoy, Completing Bethe’s Equati ons at Roots of Unity, J. Stat. Phys. 104(2001) 573–587

View on Pith 2001

[14] [14]

K. Fabricius and B. M. McCoy, Evaluation Parameters and Bethe Roots for the Six-Vertex Model at Roots of Unity, in Progress in Mathemati cal Physics Vol. 23 (MathPhys Odyssey 2001 ), edited by M. Kashiwara and T. Miwa, (Birkh¨ auser, Boston, 2002) 119–144

View on Pith 2001

[15] [15]

M. Jimbo, private communication, July 2004

View on Pith 2004

[16] [16]

G. Lusztig, Modular representations and quantum group s, Contemp. Math. 82 (1989) 59–77

View on Pith 1989

[17] [17]

G. Lusztig, Introduction to Quantum Groups (Birkh¨ auser, Boston, 1993)

View on Pith 1993

[18] [18]

V. Pasquier and H. Saleur, Common structures between ﬁn ite systems and confor- mal ﬁeld theories through quantum groups, Nucl. Phys. B 330 (1990) 523–556

View on Pith 1990

[19] [19]

P. Roche and D. Arnaudon, Irreducible representations of the quantum analogue of SU(2), Lett. Math. Phys. 17 (1989) 295–300

View on Pith 1989

[20] [20]

V. O. Tarasov, Cyclic Monodromy matrices for the R-matrix of the six-vertex model and the chiral Potts model with ﬁxed spin boundary condition s, Int. J. Mod. Phys. A 7 Suppl. 1B (1992) 963–975. 10

View on Pith 1992