Generalized Markov traces and Jucys-Murphy elements

Heorhii Zhylinskyi; Kostiantyn Tolmachov

REVIEW 2 major objections 3 minor 20 references

Generalized Markov traces and Jucys-Murphy elements

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Jucys-Murphy polynomials give all Markov traces in types B and D

desk verdict Good idea, likely fixable, but the central formula for β_n(y) has a sign error as printed; the Geck–Lambropoulou identification does not go through until it is corrected. read the letter →

arxiv 2507.19896 v1 pith:S7DFN4MJ submitted 2025-07-26 math.RT

Kostiantyn Tolmachov , Heorhii Zhylinskyi This is my paper · ORCID

classification math.RT MSC 20C0820F5557K10

keywords MarkovtracesIwahori-HeckealgebrasJucys-MurphyelementstypesBandDlinkinvariantscentralKhovanov-RozanskyhomologyHOMFLY-PTpolynomial

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

This paper proves that the general Markov traces on Iwahori-Hecke algebras of types B and D, which had previously been classified only by existence and uniqueness, are represented by explicit central elements built from commuting Jucys-Murphy elements. In type B the representing element is $\beta_n(y)=\prod_{i=1}^n(1+(y+\alpha_0)j_i^B+a^{-1}J_i^B)$; in type D it is the even-degree part of $\prod_i(1+yx_i+a^{-1}x_i^2)$ evaluated at the elements $j_i^D$. The same mechanism yields a uniform Markov trace in types A, B and D with constants $\rho=1+a$ and $\mu=v-v^{-1}$. This matters because Markov traces are what turn Hecke algebras into link invariants, and the geometric Markov traces studied via categorification arise as specializations of these formulas.

What carries the argument

The engine is the family of multiplicative Jucys-Murphy elements $J^X_n=S(X_n)^{-1}S(X_{n-1})$ in $H(X_n)$, where $S(X_n)=t_{w_0}^{-2}$ is the inverse square of the longest-element basis element. These elements commute, and any symmetric polynomial in them is central. A functional is represented by a central element $z$ when $\varphi(h)=\langle z,h\rangle$ for the natural pairing $\langle h,h'\rangle=\tau(i(h)h')$. The proof of the Markov-move properties rests on the Serre property $\langle h_1,Sh_2\rangle=\langle h_2,h_1\rangle$, which lets a factor $J^X_n=S(X_n)^{-1}S(X_{n-1})$ move across the pairing, and on the Geck-Lambropoulou uniqueness theorem, which reduces verification to normalization and to the values on $T_1\cdots T_n$ or $U_1\cdots U_{2n}$.

What would settle it

Compute, in a computer algebra system, the trace $\operatorname{tr}^B_{3,y}(T_1T_2T_3)$ using the explicit pairing $\langle\beta_3(y),T_1T_2T_3\rangle$ and compare it with $y^3$ for generic $y,v,a$ over $\mathbb{Q}(v,a)$; any deviation would falsify Theorem 3.2.2, and a similar check of $\operatorname{tr}^D_{4,y}(U_1U_2U_3U_4)=y^4$ against $\delta_4(y)$ would falsify Corollary 3.2.5.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Geck-Lambropoulou Markov traces in types B and D, uniquely determined by normalization $(1+a)^n$ and by prescribed values on $T_1\cdots T_n$ (type B) or $U_1\cdots U_{2n}$ (type D), coincide with traces written as pairings with explicit central elements. In type B the representing element is $\beta_n(y)=\prod_{i=1}^n(1+(y+\alpha_0)j_i^B+a^{-1}J_i^B)$ in $H(B_n)\otimes A(y)$, and the equality is proved by checking the defining properties of the classified trace. In type D the representing element is $\delta_n(y)$, the even-degree part of $\prod_{i=1}^n(1+yx_i+a^{-1}x_i^2)$ evaluated at the commuting elements $j_i^D$. At the specializations $y=v-v^{-1}$ (type B) and $y^2=-(v-v^{-1})^2a$ (type D), these formulas reproduce the geometric Markov traces studied in [Gom06], [WW11] and [BT22]; coefficient-wise, type B's $k$th piece is represented by $e_k(J_1^B,\ldots,J_n^B)$, and type D's by $(-1)^k e'_k(j_1^D,\ldots,j_n^D;\alpha)$, where $e'_k$ is the degree-$2k$ part of $\prod_i(1+\alpha x_i-x_i^2)$.

Load-bearing premise

The argument leans on the Geck-Lambropoulou classification, which asserts that a Markov trace in type B or D with fixed constants, normalization, and prescribed values on $T_1\cdots T_n$ (or $U_1\cdots U_{2n}$) is unique; if that theorem does not apply at the chosen specializations, the equalities identifying the explicit central elements with the known traces would not follow.

Editorial extensions

If this is right

In type B, every Markov trace with constants $\rho=1+a$, $\mu=v-v^{-1}$ and normalization $(1+a)^n$ is represented by $\beta_n(y)$, giving a closed Jucys-Murphy formula for the whole one-parameter family.
In type D, the same conclusion holds with $\delta_n(y)$, the even-degree part of $\prod_i(1+yx_i+a^{-1}x_i^2)$ evaluated at $j_i^D$.
The geometric Markov traces of [Gom06], [WW11] and [BT22] are the specializations $y=v-v^{-1}$ in type B and $y^2=-(v-v^{-1})^2a$ in type D; their coefficient-at-$a^k$ pieces are represented by $e_k(J^B)$ and $(-1)^k e'_k(j^D;\alpha)$.
The uniform trace $\operatorname{tr}_{\zeta_n}$ with $\zeta_n=\prod_i(1+a^{-1}J_i^X)$ is Markov in all classical types A, B and D, so the classical type A Markov trace of [Jon87] is a special case of the same formula.
The top coefficient of each trace is represented by the full twist $S^{-1}$, reproducing the HOMFLY-PT symmetry observed in [Kál09].

Reading between the lines

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

The same proof pattern may extend to other finite Coxeter types that carry Jucys-Murphy elements and a uniqueness theorem for Markov traces; the paper does not assert this.
The type D formula's even-degree projection suggests that odd symmetric polynomials in $j_i^B$ are invisible to traces on $H(D_n)$, which could explain why the geometric D-type trace differs from the uniform $\zeta$-trace.
If a categorification in types B and D is constructed, these explicit representatives predict filtrations of Khovanov-Rozansky homology indexed by products of Jucys-Murphy elements, as the paper's closing remark hopes for.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The short version: the paper has a good idea and the main theorem is probably true, but the central formula as printed has a sign error. The stress-test is correct: β_n(y) is defined with (y+α0), while the proof of Proposition 3.2.3 needs (y−α0). The text even prints both “y′ = y − α0” and “y′ = y + α0” in the same paragraph. With the displayed coefficient, the computation gives (y+2α0)⟨β_{n−1}, ι(h)⟩, not y⟨⋯⟩, so the identification with the Geck–Lambropoulou trace fails at the uniqueness step. That is load-bearing: Theorem 3.2.2 and Corollary 3.2.5 rest on it. The likely fix is a one-character change, but as submitted the formula is wrong.

What is genuinely new and good: the uniform ζ_n Markov trace in all classical types (Theorem 3.1.4) is a clean construction, and the proof of the Markov properties via the Kazhdan–Lusztig pairing and the Serre property is elegant. The idea of representing the Geck–Lambropoulou traces by symmetric polynomials in multiplicative Jucys–Murphy elements is natural and, as far as I know, not in the literature. The type D reduction by even/odd degree parts is also neat. The connection to the geometric traces of Gomi, Webster–Williamson, and Bezrukavnikov–Tolmachov at the specializations y=α (type B) and y^2=−aα^2 (type D) is the right kind of payoff. The citation pattern is fine; self-citations are motivational, not load-bearing.

Other soft spots are minor by comparison. Proposition 2.4.2’s displayed proof is too terse and has notation slips, but the statement is standard. The paper assumes without discussion that the bar involution extends to R=A(y) at the needed specializations; for the classification theorem that deserves a sentence or a citation. Small-n embeddings are excluded from the definition of Markov trace and handled ad hoc in Section 3.3; acceptable but should be more explicit.

Who is this for: people working on Iwahori–Hecke algebras, Markov traces, and link invariants in solid tori, and people in the Soergel-bimodule/Khovanov–Rozansky world who care about categorifying these traces. They should read it after the sign fix. I would send it to a serious referee: the construction is explicit, checkable, and worth having in the literature, and the flaw looks repairable. Just make sure the referee knows to verify the sign in β_n(y) and to ask for a cleaned-up proof of the Serre property.

Referee Report

2 major / 3 minor

Summary. The paper gives a new construction of Markov traces for Iwahori-Hecke algebras in types A, B, and D, using traces represented by central elements built from symmetric polynomials in multiplicative Jucys-Murphy elements. In type A this recovers the known construction; in types B and D the authors identify the Geck-Lambropoulou one-parameter family of Markov traces tr^X_{n,y} with traces represented by explicit central elements β_n(y) and δ_n(y) (Theorem 3.2.2 and Corollary 3.2.5). The paper then specializes the formulas to the geometric Markov traces of Gomi, Webster-Williamson, and Bezrukavnikov-Tolmachov (Section 3.3). The overall strategy is to verify the defining properties of a Markov trace and invoke the Geck-Lambropoulou uniqueness theorem, with the internal computations based on the Kazhdan-Lusztig pairing and the Serre-type property of the full-twist elements.

Significance. If the sign issue identified below is corrected, this is a genuinely useful contribution: it provides explicit polynomial formulas for Markov traces in types B and D, gives a uniform treatment across types A, B, and D, and connects the algebraic traces with known geometric constructions. The formulas are clean and likely to be valuable for further work, including possible categorification. The authors report having checked their formulas numerically in SageMath, which is a strength, although no code or output is included. The reliance on the Geck-Lambropoulou uniqueness classification is appropriate and is not circular.

major comments (2)

[Section 3.2, definition of β_n(y) and proof of Proposition 3.2.3] The displayed definition of β_n(y) uses (1 + (y + α0) j_i^B + a^{-1} J_i^B), but the proof of Proposition 3.2.3 requires the coefficient y′ to be y − α0. The text even prints both options in the sentence "y′ = y − α0, y′ = y + α0". Tracing the computation: the first summand contributes α0⟨β_{n−1}(y), ι(h)⟩, the second contributes y′⟨β_{n−1}(y), ι(h)⟩, and the total is written as (α0 + y′)⟨β_{n−1}(y), ι(h)⟩ = y⟨β_{n−1}(y), ι(h)⟩. Hence y′ must be y − α0. With the printed coefficient y + α0, the trace on T_1⋯T_n would be (y + 2α0)^n rather than y^n, so Theorem 3.2.1 no longer identifies tr_{β_n(y)} with tr^B_{n,y}. Similarly, the geometric specialization in Section 3.3 requires y − α0 = 0; with the printed sign one would obtain β_n(α) ≠ ζ_n. The definition of β_n(y) should be changed to use y − α0, and the contradictory notation for y′ should be removed.
[Section 3.2, base change of the Kazhdan-Lusztig pairing] The pairing ⟨ , ⟩ is A-antilinear in the first argument by Proposition 2.3.1(b). The displayed expansion in the proof of Proposition 3.2.3 pulls the scalar y′ out of the first argument without a conjugate, which implicitly requires ar{y}′ = y′. However, the extension of the bar involution to R = A(y) is not defined: the sentence "Extend the bar involution to R, formally writing y — we assume that such an extension exists under any specialization of y we will use" does not state what ar{y} is. If ar{y} ≠ y, the computation would acquire a conjugate on y′. Moreover, for the geometric specialization y = v − v^{-1}, the naive assignment ar{y} = y is not compatible with ar{v} = v^{-1}. The authors should specify that y is a formal variable fixed by the bar involution and explain how substitution into the resulting formulas is meant to be performed after the formal computation. This is a rigor gap in a load-bearing step, although it appears to be easily fixable.

minor comments (3)

[Section 3.2, proof of Proposition 3.2.3] In the displayed expansion, the third summand is written as a⟨β_{n−1}(y)J^B_n, ι(h)T_n⟩, but the definition of β_n(y) contains the coefficient a^{-1} for J^B_i. The term vanishes in the subsequent argument, so this does not affect the conclusion, but the coefficient should be a^{-1} for consistency.
[Section 3.3, Theorem 3.3.1] In the formula for e′_k, the symbols e_i and e_j are used without being defined. Please state that they are elementary symmetric polynomials in x_1, …, x_n, and clarify the summation convention beyond the displayed condition i + j = 2k, i, j ≥ 0.
[Section 2.3] The overline notation is used both for the Kazhdan-Lusztig involution on H(W,S) and, later, for the bar involution on the coefficient ring R. The authors should state explicitly that the latter is an extension of the former, or use a different symbol for one of the two involutions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central trace equalities are verified against an external uniqueness theorem, not assumed.

full rationale

The paper's derivation chain is non-circular. In Section 3.1, Theorem 3.1.4 constructs the uniform Markov trace tr_{zeta_n} and proves the Markov properties (U1) and (M2) directly from Proposition 2.4.2 (the Serre property of the full twist) and Corollary 2.3.2 (orthogonality of parabolic span), with no appeal to the conclusion. In Section 3.2, the elements beta_n(y) and delta_n(y) are explicit ansatze, not hidden definitions of the target traces: Proposition 3.2.3 computes tr_{beta_n(y)}(iota(h)T_n) = y tr_{beta_{n-1}(y)}(h) directly, and the proof of Theorem 3.2.2 verifies the remaining Markov axioms by reducing them to the already-proved uniform case. The identification of these candidates with the Geck-Lambropoulou traces is then made by invoking Theorem 3.2.1, which is an external classification theorem of Geck and Lambropoulou, not a self-citation of the present authors. Similarly, Corollary 3.2.5 is derived by restriction from the type-B statement using the external classification's restriction claim together with a direct parity computation. Self-citations such as [BT22] and [Tol24] appear only as motivation, perspective, or outlook and are not load-bearing in any proof. The paper does not fit any parameter to a subset of data and then rename it a prediction; the free parameter y is carried symbolically and fixed only in Section 3.3 by explicit conditions such as y - alpha_0 = 0 and y^2 = -alpha^2 a, which are stated, not imposed through an assumed equality. The apparent conflict in the printed notation 'y' = y - alpha_0, y' = y + alpha_0' and the related sign issue in beta_n(y) are correctness risks in the exposition, not circular dependencies; a typo cannot convert a verification into a definitional identity. Overall, the central claims are supported by an independent uniqueness theorem plus direct computations, so there is no circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numbers are fitted to data; a and y are formal variables in the coefficient ring. The central claim depends on standard Hecke algebra structure, the external Geck-Lambropoulou classification theorem, and an assumed extension of the bar involution. No new particles, forces, dimensions, or other entities are introduced.

assumptions (5)

domain assumption Geck-Lambropoulou classification of Markov traces in types B and D (Theorem 3.2.1)
Used to conclude that the constructed trace functionals coincide with the known traces; existence and uniqueness are cited from [GL97] and [Gec98] and not reproved in this paper.
standard math Standard structure theory of Iwahori-Hecke algebras: Kazhdan-Lusztig involution, anti-involution i, trace τ, pairing, orthogonality, and Corollary 2.3.2
Invoked throughout the proofs of Theorems 3.1.4, 3.2.2, and 3.2.5; these are standard results, for example from [GP00, Chapter 8].
standard math Serre property of the full twist element, Proposition 2.4.2
Key identity used to evaluate terms involving Jucys-Murphy elements and the full twist; the proposition is standard in the Hecke category literature, but the displayed proof in the paper appears to contain a typo.
domain assumption The bar involution extends to the coefficient ring R = A(y) and to geometric specializations
Section 3.2 states that the extension is assumed to exist; this is needed to define the one-parameter family of traces and to specialize y to geometric values in Section 3.3.
standard math The quotient and subalgebra descriptions of H(B_n) and H(D_n) inside the affine Hecke algebra (Section 2.5)
Used to define j_i^B and j_i^D and to transfer computations from type B to type D; follows from classical affine Hecke algebra theory [Lus83], [RR03].

how reviews work

0 comments

Cite this review

Pith. "Pith review of Generalized Markov traces and Jucys-Murphy elements." pith.science (2026). https://pith.science/paper/S7DFN4MJ

@misc{pith2026250719896,
  author       = {Pith},
  title        = {Pith review of: Generalized Markov traces and Jucys-Murphy elements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S7DFN4MJ}},
  note         = {Machine review of arXiv:2507.19896}
}

read the original abstract

We give a simple construction of Markov traces for Iwahori-Hecke algebras associated with infinite series of crystallographic Coxeter groups. In types B and D it is new, and generalizes a known construction in type A employing symmetric polynomials in multiplicative Jucys-Murphy elements.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 19 canonical work pages

[1]

Tilting exercises

Aleksandr Beilinson, Roman Bezrukavnikov, and Ivan Mirkovi \'c . Tilting exercises. Moscow Mathematical Journal , 4(3):547--557, 2004

work page 2004
[2]

Monodromic model for Khovanov -- Rozansky homology

Roman Bezrukavnikov and Kostiantyn Tolmachov. Monodromic model for Khovanov -- Rozansky homology. Journal f \"u r die reine und angewandte Mathematik (Crelles Journal) , 2022(787):79--124, June 2022

work page 2022
[3]

Trace functions on Iwahori-Hecke algebras

Meinolf Geck. Trace functions on Iwahori-Hecke algebras. Banach Center Publications , 42(1):87--109, 1998

work page 1998
[4]

Serre duality for Khovanov -- Rozansky homology

Eugene Gorsky, Matthew Hogancamp, Anton Mellit, and Keita Nakagane. Serre duality for Khovanov -- Rozansky homology. Selecta Mathematica , 25(5):79, 2019

work page 2019
[5]

Markov traces and knot invariants related to I wahori- H ecke algebras of type B

Meinolf Geck and Sofia Lambropoulou. Markov traces and knot invariants related to I wahori- H ecke algebras of type B . Journal f \"u r die reine und angewandte M athematik , 485:191--214, 1997

work page 1997
[6]

Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology

Eugene Gorsky, Andrei Negu t , and Jacob Rasmussen. Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology. arXiv preprint arXiv:1608.07308 , 2016

work page Pith review arXiv 2016
[7]

The markov traces and the fourier transforms

Yasushi Gomi. The markov traces and the fourier transforms. Journal of Algebra , 303(2):566--591, 2006

work page 2006
[8]

Characters of finite C oxeter groups and I wahori- H ecke algebras

Meinolf Geck and G \"o tz Pfeiffer. Characters of finite C oxeter groups and I wahori- H ecke algebras . Number 21. Oxford University Press, 2000

work page 2000

Show all 20 references

[9]

Ho and Penghui Li

Quoc P. Ho and Penghui Li. Relative serre duality for hecke categories. arXiv preprint arXiv:2504.12798 , 2025

2025
[10]

Vaughan F. R. Jones. Hecke algebra representations of braid groups and link polynomials. In New Developments In The Theory Of Knots , pages 20--73. World Scientific, 1987

1987
[11]

Meridian twisting of closed braids and the Homfly polynomial

Tam \'a s K \'a lm \'a n. Meridian twisting of closed braids and the Homfly polynomial. Mathematical Proceedings of the Cambridge Philosophical Society , 146(03):649, May 2009

2009
[12]

Triply-graded link homology and Hochschild homology of Soergel bimodules

Mikhail Khovanov. Triply-graded link homology and Hochschild homology of Soergel bimodules. International Journal of Mathematics , 18(08):869--885, 2007

2007
[13]

Solid torus links and hecke algebras of b-type

Sofia Lambropoulou. Solid torus links and hecke algebras of b-type. In Proceedings of the Conference on Quantum Topology (Manhattan, KS, 1993) , pages 225--245. World Sci. Publ. Co., Inc. River Edge, NJ, 1994

1993
[14]

Singularities, character formulas, and a q-analog of weight multiplicities

George Lusztig. Singularities, character formulas, and a q-analog of weight multiplicities. In Analyse et Topologie Sur Les Espaces Singuliers. CIRM , 6 - 10 Juillet 1981. ( Actes Du Colloque de Luminy 1981). II - III . 1983

1981
[15]

Affine braid group , JM elements and knot homology

Alexei Oblomkov and Lev Rozansky. Affine braid group , JM elements and knot homology . Transformation Groups , 24(2):531--544, June 2019

2019
[16]

Seminormal representations of weyl groups and iwahori-hecke algebras

Arun Ram. Seminormal representations of weyl groups and iwahori-hecke algebras. Proceedings of the London Mathematical Society , 75(1):99--133, 1997

1997
[17]

Affine H ecke A lgebras, C yclotomic H ecke A lgebras and C lifford T heory

Arun Ram and Jacqui Ramagge. Affine H ecke A lgebras, C yclotomic H ecke A lgebras and C lifford T heory. In A Tribute to C. S. Seshadri: Perspectives in Geometry and Representation Theory , pages 428--466, Gurgaon, 2003. Hindustan Book Agency

2003
[18]

S ageMath, the S age M athematics S oftware S ystem ( V ersion 10.6) , 2025

Sage Developers . S ageMath, the S age M athematics S oftware S ystem ( V ersion 10.6) , 2025. https://www.sagemath.org

2025
[19]

Linear structure on a finite H ecke category in type A

Kostiantyn Tolmachov. Linear structure on a finite H ecke category in type A . arXiv preprint arXiv:2403.19734 , 2024

2024
[20]

The geometry of Markov traces

Ben Webster and Geordie Williamson. The geometry of Markov traces . Duke Mathematical Journal , 160(2):401 -- 419, 2011

2011

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Tilting exercises

Aleksandr Beilinson, Roman Bezrukavnikov, and Ivan Mirkovi \'c . Tilting exercises. Moscow Mathematical Journal , 4(3):547--557, 2004

work page 2004

[2] [2]

Monodromic model for Khovanov -- Rozansky homology

Roman Bezrukavnikov and Kostiantyn Tolmachov. Monodromic model for Khovanov -- Rozansky homology. Journal f \"u r die reine und angewandte Mathematik (Crelles Journal) , 2022(787):79--124, June 2022

work page 2022

[3] [3]

Trace functions on Iwahori-Hecke algebras

Meinolf Geck. Trace functions on Iwahori-Hecke algebras. Banach Center Publications , 42(1):87--109, 1998

work page 1998

[4] [4]

Serre duality for Khovanov -- Rozansky homology

Eugene Gorsky, Matthew Hogancamp, Anton Mellit, and Keita Nakagane. Serre duality for Khovanov -- Rozansky homology. Selecta Mathematica , 25(5):79, 2019

work page 2019

[5] [5]

Markov traces and knot invariants related to I wahori- H ecke algebras of type B

Meinolf Geck and Sofia Lambropoulou. Markov traces and knot invariants related to I wahori- H ecke algebras of type B . Journal f \"u r die reine und angewandte M athematik , 485:191--214, 1997

work page 1997

[6] [6]

Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology

Eugene Gorsky, Andrei Negu t , and Jacob Rasmussen. Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology. arXiv preprint arXiv:1608.07308 , 2016

work page Pith review arXiv 2016

[7] [7]

The markov traces and the fourier transforms

Yasushi Gomi. The markov traces and the fourier transforms. Journal of Algebra , 303(2):566--591, 2006

work page 2006

[8] [8]

Characters of finite C oxeter groups and I wahori- H ecke algebras

Meinolf Geck and G \"o tz Pfeiffer. Characters of finite C oxeter groups and I wahori- H ecke algebras . Number 21. Oxford University Press, 2000

work page 2000