165:
63:
22:
1526:
1779:
2670:
different to that in
Haglund, Haiman, and Loehr's work, with many fewer terms (this formula is proved also in Macdonald's seminal work, Ch. VI (7.13)). While very useful for computation and interesting in its own right, their combinatorial formulas do not immediately imply positivity of the Kostka-Macdonald coefficients
3892:
In 1995, Macdonald introduced a non-symmetric analogue of the symmetric
Macdonald polynomials, and the symmetric Macdonald polynomials can easily be recovered from the non-symmetric counterpart. In his original definition, he shows that the non-symmetric Macdonald polynomials are a unique family of
3915:
used the exclusion process to give a direct combinatorial characterization of both symmetric and nonsymmetric
Macdonald polynomials. Their results differ from the earlier work of Haglund in part because they give a formula directly for the Macdonald polynomials rather than a transformation thereof.
1073:
is not totally ordered, and so has plenty of elements that are incomparable. Thus one must check that the corresponding polynomials are still orthogonal. The orthogonality can be proved by showing that the
Macdonald polynomials are eigenvectors for an algebra of commuting self-adjoint operators
1289:
2669:
In 2005, J. Haglund, M. Haiman and N. Loehr gave the first proof of a combinatorial interpretation of the
Macdonald polynomials. In 1988, I.G. Macdonald gave the second proof of a combinatorial interpretation of the Macdonald polynomials (equations (4.11) and (5.13)). Macdonaldâs formula is
2939:
4244:
1569:
3405:
3916:
They develop the concept of a multiline queue, which is a matrix containing balls or empty cells together with a mapping between balls and their neighbors and a combinatorial labeling mechanism. The nonsymmetric
Macdonald polynomial then satisfies:
820:
2426:
1551:
There are two other conjectures which together with the norm conjecture are collectively referred to as the
Macdonald conjectures in this context: in addition to the formula for the norm, Macdonald conjectured a formula for the value of
901:
1081:
can be chosen to vary with the length of the root, giving a three-parameter family of
Macdonald polynomials. One can also extend the definition to the nonreduced root system BC, in which case one obtains a six-parameter family (one
3547:
Note: The figure at right uses French notation for tableau, which is flipped vertically from the
English notation used on the Knowledge page for Young diagrams. French notation is more commonly used in the study of Macdonald
1521:{\displaystyle \langle P_{\lambda },P_{\lambda }\rangle =\prod _{\alpha \in R,\alpha >0}\prod _{0<i<k}{1-q^{(\lambda +k\rho ,2\alpha /(\alpha ,\alpha ))+i} \over 1-q^{(\lambda +k\rho ,2\alpha /(\alpha ,\alpha ))-i}}.}
2200:
1003:
3999:
3789:
3030:
703:
2793:
4081:
3868:
4665:
Dean
Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, NJ. University Lecture Series, 12. American Mathematical Society, Providence, RI, 1998. xvi+53 pp.
1774:{\displaystyle {\frac {P_{\lambda }(\dots ,q^{\mu _{i}}t^{\rho _{i}},\dots )}{P_{\lambda }(t^{\rho })}}={\frac {P_{\mu }(\dots ,q^{\lambda _{i}}t^{\rho _{i}},\dots )}{P_{\mu }(t^{\rho })}}.}
3607:
3109:
2083:
2956:
are certain combinatorial statistics (functions) defined on the filling Ï. This formula expresses the Macdonald polynomials in infinitely many variables. To obtain the polynomials in
2758:
1860:
276:
in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald originally associated his polynomials with weights λ of finite root systems and used just one variable
2716:
3244:
3236:
3188:
1901:
714:
482:
2231:
elements of the diagram of the partition Ό, regarded as a subset of the pairs of non-negative integers. For example, if Ό is the partition 3 = 2 + 1 of
4073:
3047:
508:
3136:
2785:
2624:
1931:
3437:
2590:
2549:
2508:
2467:
4381:
Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN 0-19-853489-2
4028:
4287:
4048:
3634:
3457:
2266:
4267:
3515:
3486:
826:
3543:
2611:
4526:
Symmetric functions 2001: surveys of developments and perspectives, 1–64, NATO Sci. Ser. II Math. Phys. Chem., 74, Kluwer Acad. Publ., Dordrecht, 2002.
3050:
This depicts the arm and the leg of a square of a Young diagram. The arm is the number of squares to its right, and the leg is the number of squares above it.
1051:. The existence of polynomials with these properties is easy to show (for any inner product). A key property of the Macdonald polynomials is that they are
1094:. It is sometimes better to regard Macdonald polynomials as depending on a possibly non-reduced affine root system. In this case, there is one parameter
2094:
84:
77:
4645:
Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp.
1964:
with non-negative integer coefficients. These conjectures are now proved; the hardest and final step was proving the positivity, which was done by
939:
3922:
3642:
4308:
4394:
Corteel, Sylvie; Mandelshtam, Olya; Williams, Lauren (2018), "From multiline queues to Macdonald polynomials via the exclusion process",
3893:
polynomials orthogonal to a certain inner product, as well as satisfying a triangularity property when expanded in the monomial basis.
4691:
127:
2971:
2934:{\displaystyle {\widetilde {H}}_{\mu }(x;q,t)=\sum _{\sigma :\mu \to \mathbb {Z} _{+}}q^{inv(\sigma )}t^{maj(\sigma )}x^{\sigma }}
2654:. This immediately implies the Macdonald positivity conjecture because character multiplicities have to be non-negative integers.
99:
2657:
Ian Grojnowski and Mark Haiman found another proof of the Macdonald positivity conjecture by proving a positivity conjecture for
625:
4239:{\displaystyle P_{\lambda }({\textbf {x}};q,t)=\sum _{\mu }E_{\mu }(x_{1},...,x_{n};q,t)=\sum _{\mu }\sum _{Q}\mathrm {wt} (Q)}
2718:
as the give the decomposition of the Macdonald polynomials into monomial symmetric functions rather than into Schur functions.
35:
3899:
The non-symmetric Macdonald polynomials specialize to Demazure characters by taking q=t=0, and to key polynomials when q=t=â.
4730:
106:
3800:
1123:
of the representations of the compact group of the root system, or the Schur functions in the case of root systems of type
2647:! conjecture implied that the Kostka–Macdonald coefficients were graded character multiplicities for the modules
113:
4670:
4650:
226:
208:
186:
146:
49:
4075:
is a weighting function mapping those queues to specific polynomials. The symmetric Macdonald polynomial satisfies:
3555:
3057:
179:
4777:
1074:
with 1-dimensional eigenspaces, and using the fact that eigenspaces for different eigenvalues must be orthogonal.
95:
4787:
3912:
1143:
2015:
455:
336:
is the rank of the affine root system. They generalize many other families of orthogonal polynomials, such as
341:
3896:
In 2007, Haglund, Haiman and Loehr gave a combinatorial formula for the non-symmetric Macdonald polynomials.
2727:
1956:-Kostka coefficients. Macdonald conjectured that the Kostka–Macdonald coefficients were polynomials in
1829:
1189:
3400:{\displaystyle J_{\lambda }(x;q,t)=\prod _{s\in D(\lambda )}(1-q^{a(s)}t^{1+l(s)})\cdot P_{\lambda }(x;q,t)}
2673:
1232:, which in turn include as special cases most of the named families of orthogonal polynomials in 1 variable.
1229:
345:
1796:, with the extension to the BC case following shortly thereafter via work of van Diejen, Noumi, and Sahi.
815:{\displaystyle \Delta =\prod _{\alpha \in R}{(e^{\alpha };q)_{\infty } \over (te^{\alpha };q)_{\infty }}.}
4782:
4725:, Cambridge Tracts in Mathematics, vol. 157, Cambridge: Cambridge University Press, pp. x+175,
4512:
3193:
3145:
1793:
1536:
511:
402:
1869:
461:
41:
4579:
1978:
It is still a central open problem in algebraic combinatorics to find a combinatorial formula for the
1539:. The conjecture had previously been proved case-by-case for all roots systems except those of type
4053:
1135:
487:
173:
120:
3114:
2763:
1909:
356:
are Macdonald polynomials of certain non-reduced root systems. They have deep relationships with
3874:
3413:
2555:
2514:
2473:
2432:
1259:
1091:
353:
73:
4306:
Haglund, J.; Haiman, M.; Loehr, N. (2005), "A combinatorial formula for Macdonald polynomials",
2421:{\displaystyle \Delta _{\mu }=x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{1}y_{3}}
1047:
In other words, the Macdonald polynomials are obtained by orthogonalizing the obvious basis for
4584:
4007:
266:
190:
4521:
4272:
4033:
3612:
3442:
4420:
1069:ă = 0 whenever λ â ÎŒ. This is not a trivial consequence of the definition because
4740:
4714:
4676:
4656:
4635:
4530:
4473:
4349:
896:{\displaystyle \langle f,g\rangle =({\text{constant term of }}f{\overline {g}}\Delta )/|W|}
357:
269:
4252:
3491:
3462:
8:
2960:
variables, simply restrict the formula to fillings that only use the integers 1, 2, ...,
706:
4477:
4365:
Publ. I.R.M.A. Strasbourg, 1988, 372/Sâ20 Actes 20e SĂ©minaire Lotharingien, p. 131â171.
310:
orbits of roots in the affine root system. The Macdonald polynomials are polynomials in
4700:
4567:
4549:
4445:
4395:
4317:
3528:
2632:
2596:
1203:
365:
281:
4496:
4726:
4689:
Macdonald, I. G. (2000â2001), "Orthogonal polynomials associated with root systems",
4666:
4646:
4623:
4501:
4457:
4437:
4337:
4418:
Cherednik, Ivan (1995), "Double Affine Hecke Algebras and Macdonald's Conjectures",
4382:
1535:, and proved for all (reduced) root systems by Cherednik (1995) using properties of
4615:
4593:
4571:
4559:
4491:
4481:
4429:
4327:
1904:
1532:
273:
4598:
4563:
4332:
2195:{\displaystyle \Delta _{\mu }=\det(x_{i}^{p_{j}}y_{i}^{q_{j}})_{1\leq i,j,\leq n}}
4736:
4710:
4673:
4653:
4631:
4527:
4345:
2636:
1181:
1161:
orbits, which are the monomial symmetric functions when the root system has type
337:
2639:). Earlier results of Haiman and Garsia had already shown that this implied the
1098:
associated to each orbit of roots in the affine root system, plus one parameter
3908:
2658:
1998:
1785:
1120:
440:
361:
998:{\displaystyle P_{\lambda }=\sum _{\mu \leq \lambda }u_{\lambda \mu }m_{\mu }}
4771:
4627:
4441:
4341:
3881:
2945:
1991:
1969:
4486:
4505:
406:
3994:{\displaystyle E_{\lambda }({\textbf {x}};q,t)=\sum _{Q}\mathrm {wt} (Q)}
2002:
1965:
1866:
below) form an orthogonal basis of the space of symmetric functions over
1077:
In the case of non-simply-laced root systems (B, C, F, G), the parameter
528:
388:
240:
4705:
4554:
3902:
3784:{\displaystyle {\widetilde {H}}_{\mu }(x;q,t)=t^{-n(\mu )}J_{\mu }\left}
3111:
in the formula above are related to the classical Macdonald polynomials
1863:
4449:
556:
416:
369:
4517:
Current Developments in Mathematics 2002, no. 1 (2002), 39–111.
2664:
4761:
4322:
280:, but later realized that it is more natural to associate them with
4754:
4619:
4538:
Hilbert schemes, polygraphs, and the Macdonald positivity conjecture
4433:
1531:
This was conjectured by Macdonald (1982) as a generalization of the
62:
4523:
Notes on Macdonald polynomials and the geometry of Hilbert schemes.
4400:
531:: half the sum of the positive roots; this is a special element of
3880:
This formula can be used to prove Knop and Sahi's formula for the
612:
is an orbit sum; these elements form a basis for the subalgebra
4580:"Lectures on affine Hecke algebras and Macdonald's conjectures"
4366:
592:, and this is extended by linearity to the whole group algebra.
4606:
Macdonald, I. G. (1982), "Some conjectures for root systems",
2619:
Haiman's proof of the Macdonald positivity conjecture and the
1814:
the Macdonald polynomials are simply symmetric polynomials in
3025:{\displaystyle x_{1}^{\sigma _{1}}x_{2}^{\sigma _{2}}\cdots }
1784:
Again, these were proved for general reduced root systems by
541:
is a field of characteristic 0, usually the rational numbers.
284:
rather than finite root systems, in which case the variable
2253:) are (0, 0), (0, 1), (1, 0), and the space
1818:
variables with coefficients that are rational functions of
4393:
698:{\displaystyle (a;q)_{\infty }=\prod _{r\geq 0}(1-aq^{r})}
3046:
1283:, then the norm of the Macdonald polynomials is given by
1266:
3039:
is the number of boxes in the filling of Ό with content
4514:
Combinatorics, symmetric functions, and Hilbert schemes
1209:
For the non-reduced rank 1 affine root system of type (
4249:
where the outer sum is over all distinct compositions
3887:
1799:
1102:. The number of orbits of roots can vary from 1 to 5.
933:
are uniquely defined by the following two conditions:
4275:
4255:
4084:
4056:
4036:
4010:
3925:
3903:
Combinatorial formulae based on the exclusion process
3863:{\displaystyle n(\mu )=\sum _{i}\mu _{i}\cdot (i-1).}
3803:
3645:
3615:
3558:
3531:
3494:
3465:
3445:
3439:
is the collection of squares in the Young diagram of
3416:
3247:
3196:
3148:
3117:
3060:
2974:
2796:
2766:
2730:
2676:
2599:
2558:
2517:
2476:
2435:
2269:
2097:
2018:
1912:
1872:
1832:
1572:
1292:
942:
829:
717:
628:
490:
464:
4462:
Breakthroughs in the theory of Macdonald polynomials
1134:= 0 the Macdonald polynomials become the (rescaled)
2665:
Combinatorial formula for the Macdonald polynomials
4305:
4281:
4261:
4238:
4067:
4042:
4022:
3993:
3862:
3783:
3628:
3601:
3537:
3509:
3480:
3451:
3431:
3399:
3238:that clears the denominators of the coefficients:
3230:
3182:
3130:
3103:
3024:
2933:
2779:
2752:
2710:
2605:
2584:
2543:
2502:
2461:
2420:
2194:
2077:
1925:
1895:
1854:
1773:
1520:
1157:=1 the Macdonald polynomials become the sums over
997:
895:
814:
697:
502:
476:
4683:Affine Hecke algebras and orthogonal polynomials.
1235:For the non-reduced affine root system of type (
4769:
4723:Affine Hecke algebras and orthogonal polynomials
2111:
520:is the set of dominant weights: the elements of
4663:Symmetric functions and orthogonal polynomials.
3138:via a sequence of transformations. First, the
1043:are orthogonal if λ < μ.
288:can be replaced by several different variables
4688:
3602:{\displaystyle {\widetilde {H}}_{\mu }(x;q,t)}
3104:{\displaystyle {\widetilde {H}}_{\mu }(x;q,t)}
2088:spanned by all higher partial derivatives of
1903:, and therefore can be expressed in terms of
535:in the interior of the positive Weyl chamber.
4455:
4309:Journal of the American Mathematical Society
1319:
1293:
842:
830:
1180:tend to 1 the Macdonald polynomials become
50:Learn how and when to remove these messages
2078:{\displaystyle D_{\mu }=C\,\Delta _{\mu }}
348:, which in turn include most of the named
4720:
4704:
4643:Symmetric functions and Hall polynomials.
4605:
4597:
4553:
4495:
4485:
4417:
4399:
4379:Symmetric functions and Hall polynomials.
4331:
4321:
2859:
2064:
1874:
1789:
352:orthogonal polynomials as special cases.
227:Learn how and when to remove this message
209:Learn how and when to remove this message
147:Learn how and when to remove this message
4577:
4355:
3045:
903:is the inner product of two elements of
172:This article includes a list of general
2753:{\displaystyle {\widetilde {H}}_{\mu }}
2623:! conjecture involved showing that the
1855:{\displaystyle {\widetilde {H}}_{\mu }}
510:is a nonnegative linear combination of
4770:
4692:SĂ©minaire Lotharingien de Combinatoire
4535:
4387:
4371:
4299:
3552:The transformed Macdonald polynomials
3142:of the Macdonald polynomials, denoted
3054:The transformed Macdonald polynomials
2711:{\displaystyle K_{\lambda \mu }(q,t),}
2616:which has dimension 6 = 3!.
1267:The Macdonald constant term conjecture
83:Please improve this article by adding
4608:SIAM Journal on Mathematical Analysis
4428:(1), Annals of Mathematics: 191â216,
1804:In the case of roots systems of type
1258:), the Macdonald polynomials are the
1228:), the Macdonald polynomials are the
1119:the Macdonald polynomials become the
3609:can then be defined in terms of the
2005:states that for each partition Ό of
1202:, the Macdonald polynomials are the
272:in several variables, introduced by
158:
56:
15:
4363:A new class of symmetric functions.
4100:
3941:
3888:Non-symmetric Macdonald polynomials
3873:The bracket notation above denotes
3231:{\displaystyle P_{\lambda }(x;q,t)}
3183:{\displaystyle J_{\lambda }(x;q,t)}
1950:Kostka–Macdonald coefficients
1800:The Macdonald positivity conjecture
563:, with a basis of elements written
433:(the lattice spanned by the roots).
13:
4289:, and the inner sum is as before.
4223:
4220:
4061:
4058:
3978:
3975:
2271:
2099:
2066:
2052:
2039:
1862:of the Macdonald polynomials (see
869:
801:
767:
718:
646:
405:, to which corresponds a positive
178:it lacks sufficient corresponding
14:
4799:
4747:
2723:transformed Macdonald polynomials
1896:{\displaystyle \mathbb {Q} (q,t)}
477:{\displaystyle \mu \leq \lambda }
31:This article has multiple issues.
1985:
1948:) of these relations are called
1826:. A certain transformed version
1184:when the root system is of type
163:
61:
20:
4411:
2235: = 3 then the pairs (
911:is a positive integer power of
39:or discuss these issues on the
4685:SĂ©minaire Bourbaki 797 (1995).
4233:
4227:
4193:
4143:
4117:
4095:
3988:
3982:
3958:
3936:
3854:
3842:
3813:
3807:
3706:
3700:
3683:
3665:
3596:
3578:
3504:
3498:
3475:
3469:
3426:
3420:
3394:
3376:
3360:
3355:
3349:
3330:
3324:
3307:
3302:
3296:
3276:
3258:
3225:
3207:
3177:
3159:
3098:
3080:
2916:
2910:
2891:
2885:
2854:
2834:
2816:
2702:
2690:
2162:
2114:
2061:
2035:
1890:
1878:
1762:
1749:
1734:
1685:
1663:
1650:
1635:
1586:
1501:
1498:
1486:
1457:
1433:
1430:
1418:
1389:
1192:for more general root systems.
1146:when the root system has type
1086:for each orbit of roots, plus
889:
881:
872:
848:
797:
774:
763:
743:
692:
670:
642:
629:
372:made by Macdonald about them.
1:
4599:10.1090/S0273-0979-97-00727-1
4564:10.1090/S0894-0347-01-00373-3
4333:10.1090/S0894-0347-05-00485-6
4292:
4068:{\displaystyle \mathrm {wt} }
524:in the positive Weyl chamber.
503:{\displaystyle \lambda -\mu }
375:
85:secondary or tertiary sources
4764:about Macdonald polynomials.
3131:{\displaystyle P_{\lambda }}
2944:where Ï is a filling of the
2780:{\displaystyle P_{\lambda }}
1926:{\displaystyle s_{\lambda }}
1794:double affine Hecke algebras
1537:double affine Hecke algebras
864:
707:infinite q-Pochhammer symbol
7:
4753:Mike Zabrocki's page about
3545:, as shown in the figure.
3432:{\displaystyle D(\lambda )}
2643:! conjecture, and that the
2585:{\displaystyle x_{1}-x_{3}}
2544:{\displaystyle x_{3}-x_{2}}
2503:{\displaystyle y_{3}-y_{1}}
2462:{\displaystyle y_{2}-y_{3}}
1195:For the affine root system
1144:HallâLittlewood polynomials
1105:
342:HallâLittlewood polynomials
10:
4804:
4269:which are permutations of
4004:where the sum is over all
2625:isospectral Hilbert scheme
1279:for some positive integer
1012:is a rational function of
4721:Macdonald, I. G. (2003),
4030:multiline queues of type
4023:{\displaystyle L\times n}
2968:should be interpreted as
1190:HeckmanâOpdam polynomials
1136:zonal spherical functions
380:First fix some notation:
4578:Kirillov, A. A. (1997),
4282:{\displaystyle \lambda }
4043:{\displaystyle \lambda }
3629:{\displaystyle J_{\mu }}
3452:{\displaystyle \lambda }
1230:AskeyâWilson polynomials
346:AskeyâWilson polynomials
4778:Algebraic combinatorics
4487:10.1073/pnas.0409705102
3875:plethystic substitution
1968:(2001), by proving the
1260:Koornwinder polynomials
1092:Koornwinder polynomials
456:ordering on the weights
429:is the root lattice of
391:in a real vector space
354:Koornwinder polynomials
306:), one for each of the
193:more precise citations.
96:"Macdonald polynomials"
4788:Orthogonal polynomials
4585:Bull. Amer. Math. Soc.
4536:Haiman, Mark (2001), "
4283:
4263:
4240:
4069:
4044:
4024:
3995:
3911:, O. Mandelshtam, and
3864:
3785:
3630:
3603:
3539:
3511:
3482:
3453:
3433:
3401:
3232:
3184:
3132:
3105:
3051:
3026:
2935:
2781:
2760:rather than the usual
2754:
2712:
2631:points in a plane was
2607:
2586:
2545:
2504:
2463:
2422:
2196:
2079:
1982:-Kostka coefficients.
1927:
1897:
1856:
1775:
1522:
999:
897:
853:constant term of
816:
699:
504:
478:
72:relies excessively on
4755:Macdonald polynomials
4421:Annals of Mathematics
4284:
4264:
4241:
4070:
4045:
4025:
3996:
3865:
3786:
3631:
3604:
3540:
3512:
3483:
3454:
3434:
3402:
3233:
3190:, is a re-scaling of
3185:
3133:
3106:
3049:
3027:
2936:
2782:
2755:
2713:
2608:
2587:
2546:
2505:
2464:
2423:
2197:
2080:
1928:
1898:
1864:Combinatorial formula
1857:
1776:
1523:
1000:
921:Macdonald polynomials
898:
817:
700:
616:of elements fixed by
505:
479:
364:, which were used to
358:affine Hecke algebras
270:symmetric polynomials
245:Macdonald polynomials
4273:
4262:{\displaystyle \mu }
4253:
4082:
4054:
4034:
4008:
3923:
3801:
3643:
3613:
3556:
3529:
3510:{\displaystyle l(s)}
3492:
3481:{\displaystyle a(s)}
3463:
3443:
3414:
3245:
3194:
3146:
3115:
3058:
2972:
2794:
2764:
2728:
2674:
2633:Cohen–Macaulay
2597:
2556:
2515:
2474:
2433:
2267:
2095:
2016:
1933:. The coefficients
1910:
1870:
1830:
1570:
1548:by several authors.
1290:
1142:-adic group, or the
940:
827:
715:
626:
488:
462:
4542:J. Amer. Math. Soc.
4478:2005PNAS..102.3891G
4460:(March 15, 2005), "
3018:
2996:
2160:
2138:
282:affine root systems
265:) are a family of
4783:Algebraic geometry
4458:Remmel, Jeffrey B.
4279:
4259:
4236:
4218:
4208:
4132:
4065:
4040:
4020:
3991:
3973:
3860:
3828:
3781:
3626:
3599:
3535:
3507:
3478:
3449:
3429:
3397:
3306:
3228:
3180:
3128:
3101:
3052:
3022:
2997:
2975:
2931:
2870:
2777:
2750:
2708:
2603:
2582:
2541:
2500:
2459:
2418:
2227:) run through the
2192:
2139:
2117:
2075:
1923:
1893:
1852:
1771:
1518:
1374:
1352:
1204:Rogers polynomials
995:
971:
893:
812:
739:
695:
669:
500:
474:
4732:978-0-521-82472-9
4681:Macdonald, I. G.
4661:Macdonald, I. G.
4641:Macdonald, I. G.
4472:(11): 3891â3894,
4456:Garsia, Adriano;
4424:, Second Series,
4377:Macdonald, I. G.
4361:Macdonald, I. G.
4209:
4199:
4123:
4102:
3964:
3943:
3819:
3752:
3656:
3569:
3538:{\displaystyle s}
3282:
3071:
2840:
2807:
2741:
2606:{\displaystyle 1}
1995:! conjecture
1973:! conjecture
1843:
1766:
1667:
1563:, and a symmetry
1513:
1353:
1325:
1138:for a semisimple
956:
867:
854:
807:
724:
654:
237:
236:
229:
219:
218:
211:
157:
156:
149:
131:
54:
4795:
4743:
4717:
4708:
4638:
4602:
4601:
4574:
4557:
4508:
4499:
4489:
4452:
4405:
4404:
4403:
4391:
4385:
4375:
4369:
4359:
4353:
4352:
4335:
4325:
4303:
4288:
4286:
4285:
4280:
4268:
4266:
4265:
4260:
4245:
4243:
4242:
4237:
4226:
4217:
4207:
4180:
4179:
4155:
4154:
4142:
4141:
4131:
4104:
4103:
4094:
4093:
4074:
4072:
4071:
4066:
4064:
4049:
4047:
4046:
4041:
4029:
4027:
4026:
4021:
4000:
3998:
3997:
3992:
3981:
3972:
3945:
3944:
3935:
3934:
3882:Jack polynomials
3869:
3867:
3866:
3861:
3838:
3837:
3827:
3790:
3788:
3787:
3782:
3780:
3776:
3775:
3774:
3753:
3751:
3750:
3749:
3727:
3720:
3719:
3710:
3709:
3664:
3663:
3658:
3657:
3649:
3635:
3633:
3632:
3627:
3625:
3624:
3608:
3606:
3605:
3600:
3577:
3576:
3571:
3570:
3562:
3544:
3542:
3541:
3536:
3516:
3514:
3513:
3508:
3487:
3485:
3484:
3479:
3458:
3456:
3455:
3450:
3438:
3436:
3435:
3430:
3406:
3404:
3403:
3398:
3375:
3374:
3359:
3358:
3334:
3333:
3305:
3257:
3256:
3237:
3235:
3234:
3229:
3206:
3205:
3189:
3187:
3186:
3181:
3158:
3157:
3137:
3135:
3134:
3129:
3127:
3126:
3110:
3108:
3107:
3102:
3079:
3078:
3073:
3072:
3064:
3031:
3029:
3028:
3023:
3017:
3016:
3015:
3005:
2995:
2994:
2993:
2983:
2940:
2938:
2937:
2932:
2930:
2929:
2920:
2919:
2895:
2894:
2869:
2868:
2867:
2862:
2815:
2814:
2809:
2808:
2800:
2786:
2784:
2783:
2778:
2776:
2775:
2759:
2757:
2756:
2751:
2749:
2748:
2743:
2742:
2734:
2717:
2715:
2714:
2709:
2689:
2688:
2612:
2610:
2609:
2604:
2591:
2589:
2588:
2583:
2581:
2580:
2568:
2567:
2550:
2548:
2547:
2542:
2540:
2539:
2527:
2526:
2509:
2507:
2506:
2501:
2499:
2498:
2486:
2485:
2468:
2466:
2465:
2460:
2458:
2457:
2445:
2444:
2427:
2425:
2424:
2419:
2417:
2416:
2407:
2406:
2394:
2393:
2384:
2383:
2371:
2370:
2361:
2360:
2348:
2347:
2338:
2337:
2325:
2324:
2315:
2314:
2302:
2301:
2292:
2291:
2279:
2278:
2201:
2199:
2198:
2193:
2191:
2190:
2159:
2158:
2157:
2147:
2137:
2136:
2135:
2125:
2107:
2106:
2084:
2082:
2081:
2076:
2074:
2073:
2060:
2059:
2047:
2046:
2028:
2027:
1932:
1930:
1929:
1924:
1922:
1921:
1902:
1900:
1899:
1894:
1877:
1861:
1859:
1858:
1853:
1851:
1850:
1845:
1844:
1836:
1780:
1778:
1777:
1772:
1767:
1765:
1761:
1760:
1748:
1747:
1737:
1727:
1726:
1725:
1724:
1710:
1709:
1708:
1707:
1684:
1683:
1673:
1668:
1666:
1662:
1661:
1649:
1648:
1638:
1628:
1627:
1626:
1625:
1611:
1610:
1609:
1608:
1585:
1584:
1574:
1533:Dyson conjecture
1527:
1525:
1524:
1519:
1514:
1512:
1511:
1510:
1485:
1444:
1443:
1442:
1417:
1376:
1373:
1351:
1318:
1317:
1305:
1304:
1248:
1247:
1220:
1219:
1182:Jack polynomials
1004:
1002:
1001:
996:
994:
993:
984:
983:
970:
952:
951:
907:, at least when
902:
900:
899:
894:
892:
884:
879:
868:
860:
855:
852:
821:
819:
818:
813:
808:
806:
805:
804:
789:
788:
772:
771:
770:
755:
754:
741:
738:
704:
702:
701:
696:
691:
690:
668:
650:
649:
587:
509:
507:
506:
501:
483:
481:
480:
475:
351:
338:Jack polynomials
232:
225:
214:
207:
203:
200:
194:
189:this article by
180:inline citations
167:
166:
159:
152:
145:
141:
138:
132:
130:
89:
65:
57:
46:
24:
23:
16:
4803:
4802:
4798:
4797:
4796:
4794:
4793:
4792:
4768:
4767:
4762:Haiman's papers
4750:
4733:
4706:math.QA/0011046
4620:10.1137/0513070
4614:(6): 988â1007,
4555:math.AG/0010246
4548:(4): 941â1006,
4434:10.2307/2118632
4414:
4409:
4408:
4392:
4388:
4376:
4372:
4360:
4356:
4304:
4300:
4295:
4274:
4271:
4270:
4254:
4251:
4250:
4219:
4213:
4203:
4175:
4171:
4150:
4146:
4137:
4133:
4127:
4099:
4098:
4089:
4085:
4083:
4080:
4079:
4057:
4055:
4052:
4051:
4035:
4032:
4031:
4009:
4006:
4005:
3974:
3968:
3940:
3939:
3930:
3926:
3924:
3921:
3920:
3905:
3890:
3833:
3829:
3823:
3802:
3799:
3798:
3767:
3763:
3742:
3738:
3731:
3726:
3725:
3721:
3715:
3711:
3693:
3689:
3659:
3648:
3647:
3646:
3644:
3641:
3640:
3620:
3616:
3614:
3611:
3610:
3572:
3561:
3560:
3559:
3557:
3554:
3553:
3530:
3527:
3526:
3493:
3490:
3489:
3464:
3461:
3460:
3444:
3441:
3440:
3415:
3412:
3411:
3370:
3366:
3339:
3335:
3320:
3316:
3286:
3252:
3248:
3246:
3243:
3242:
3201:
3197:
3195:
3192:
3191:
3153:
3149:
3147:
3144:
3143:
3122:
3118:
3116:
3113:
3112:
3074:
3063:
3062:
3061:
3059:
3056:
3055:
3037:
3011:
3007:
3006:
3001:
2989:
2985:
2984:
2979:
2973:
2970:
2969:
2925:
2921:
2900:
2896:
2875:
2871:
2863:
2858:
2857:
2844:
2810:
2799:
2798:
2797:
2795:
2792:
2791:
2771:
2767:
2765:
2762:
2761:
2744:
2733:
2732:
2731:
2729:
2726:
2725:
2721:Written in the
2681:
2677:
2675:
2672:
2671:
2667:
2659:LLT polynomials
2653:
2598:
2595:
2594:
2576:
2572:
2563:
2559:
2557:
2554:
2553:
2535:
2531:
2522:
2518:
2516:
2513:
2512:
2494:
2490:
2481:
2477:
2475:
2472:
2471:
2453:
2449:
2440:
2436:
2434:
2431:
2430:
2412:
2408:
2402:
2398:
2389:
2385:
2379:
2375:
2366:
2362:
2356:
2352:
2343:
2339:
2333:
2329:
2320:
2316:
2310:
2306:
2297:
2293:
2287:
2283:
2274:
2270:
2268:
2265:
2264:
2259:
2252:
2243:
2226:
2217:
2165:
2161:
2153:
2149:
2148:
2143:
2131:
2127:
2126:
2121:
2102:
2098:
2096:
2093:
2092:
2069:
2065:
2055:
2051:
2042:
2038:
2023:
2019:
2017:
2014:
2013:
1988:
1939:
1917:
1913:
1911:
1908:
1907:
1905:Schur functions
1873:
1871:
1868:
1867:
1846:
1835:
1834:
1833:
1831:
1828:
1827:
1813:
1802:
1756:
1752:
1743:
1739:
1738:
1720:
1716:
1715:
1711:
1703:
1699:
1698:
1694:
1679:
1675:
1674:
1672:
1657:
1653:
1644:
1640:
1639:
1621:
1617:
1616:
1612:
1604:
1600:
1599:
1595:
1580:
1576:
1575:
1573:
1571:
1568:
1567:
1558:
1547:
1481:
1456:
1452:
1445:
1413:
1388:
1384:
1377:
1375:
1357:
1329:
1313:
1309:
1300:
1296:
1291:
1288:
1287:
1269:
1257:
1246:
1241:
1240:
1239:
1227:
1218:
1215:
1214:
1213:
1201:
1121:Weyl characters
1108:
1068:
1061:
1042:
1035:
1026:
1011:
989:
985:
976:
972:
960:
947:
943:
941:
938:
937:
928:
888:
880:
875:
859:
851:
828:
825:
824:
800:
796:
784:
780:
773:
766:
762:
750:
746:
742:
740:
728:
716:
713:
712:
686:
682:
658:
645:
641:
627:
624:
623:
608:
600:
583:
489:
486:
485:
484:if and only if
463:
460:
459:
401:is a choice of
378:
362:Hilbert schemes
349:
330:
324:
304:
298:
252:
233:
222:
221:
220:
215:
204:
198:
195:
185:Please help to
184:
168:
164:
153:
142:
136:
133:
90:
88:
82:
78:primary sources
66:
25:
21:
12:
11:
5:
4801:
4791:
4790:
4785:
4780:
4766:
4765:
4758:
4749:
4748:External links
4746:
4745:
4744:
4731:
4718:
4686:
4679:
4659:
4639:
4603:
4592:(3): 251â292,
4575:
4533:
4518:
4509:
4453:
4413:
4410:
4407:
4406:
4386:
4370:
4354:
4316:(3): 735â761,
4297:
4296:
4294:
4291:
4278:
4258:
4247:
4246:
4235:
4232:
4229:
4225:
4222:
4216:
4212:
4206:
4202:
4198:
4195:
4192:
4189:
4186:
4183:
4178:
4174:
4170:
4167:
4164:
4161:
4158:
4153:
4149:
4145:
4140:
4136:
4130:
4126:
4122:
4119:
4116:
4113:
4110:
4107:
4097:
4092:
4088:
4063:
4060:
4039:
4019:
4016:
4013:
4002:
4001:
3990:
3987:
3984:
3980:
3977:
3971:
3967:
3963:
3960:
3957:
3954:
3951:
3948:
3938:
3933:
3929:
3904:
3901:
3889:
3886:
3871:
3870:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3836:
3832:
3826:
3822:
3818:
3815:
3812:
3809:
3806:
3792:
3791:
3779:
3773:
3770:
3766:
3762:
3759:
3756:
3748:
3745:
3741:
3737:
3734:
3730:
3724:
3718:
3714:
3708:
3705:
3702:
3699:
3696:
3692:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3662:
3655:
3652:
3623:
3619:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3575:
3568:
3565:
3534:
3525:of the square
3506:
3503:
3500:
3497:
3477:
3474:
3471:
3468:
3448:
3428:
3425:
3422:
3419:
3408:
3407:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3373:
3369:
3365:
3362:
3357:
3354:
3351:
3348:
3345:
3342:
3338:
3332:
3329:
3326:
3323:
3319:
3315:
3312:
3309:
3304:
3301:
3298:
3295:
3292:
3289:
3285:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3255:
3251:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3204:
3200:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3156:
3152:
3125:
3121:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3077:
3070:
3067:
3035:
3021:
3014:
3010:
3004:
3000:
2992:
2988:
2982:
2978:
2942:
2941:
2928:
2924:
2918:
2915:
2912:
2909:
2906:
2903:
2899:
2893:
2890:
2887:
2884:
2881:
2878:
2874:
2866:
2861:
2856:
2853:
2850:
2847:
2843:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2813:
2806:
2803:
2774:
2770:
2747:
2740:
2737:
2707:
2704:
2701:
2698:
2695:
2692:
2687:
2684:
2680:
2666:
2663:
2651:
2614:
2613:
2602:
2592:
2579:
2575:
2571:
2566:
2562:
2551:
2538:
2534:
2530:
2525:
2521:
2510:
2497:
2493:
2489:
2484:
2480:
2469:
2456:
2452:
2448:
2443:
2439:
2428:
2415:
2411:
2405:
2401:
2397:
2392:
2388:
2382:
2378:
2374:
2369:
2365:
2359:
2355:
2351:
2346:
2342:
2336:
2332:
2328:
2323:
2319:
2313:
2309:
2305:
2300:
2296:
2290:
2286:
2282:
2277:
2273:
2260:is spanned by
2257:
2248:
2239:
2222:
2213:
2205:has dimension
2203:
2202:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2164:
2156:
2152:
2146:
2142:
2134:
2130:
2124:
2120:
2116:
2113:
2110:
2105:
2101:
2086:
2085:
2072:
2068:
2063:
2058:
2054:
2050:
2045:
2041:
2037:
2034:
2031:
2026:
2022:
1999:Adriano Garsia
1987:
1984:
1937:
1920:
1916:
1892:
1889:
1886:
1883:
1880:
1876:
1849:
1842:
1839:
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1798:
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1770:
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1759:
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1706:
1702:
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1474:
1471:
1468:
1465:
1462:
1459:
1455:
1451:
1448:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1416:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1387:
1383:
1380:
1372:
1369:
1366:
1363:
1360:
1356:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1328:
1324:
1321:
1316:
1312:
1308:
1303:
1299:
1295:
1268:
1265:
1264:
1263:
1253:
1242:
1233:
1225:
1216:
1207:
1199:
1193:
1166:
1151:
1128:
1107:
1104:
1066:
1059:
1045:
1044:
1040:
1033:
1028:
1024:
1009:
992:
988:
982:
979:
975:
969:
966:
963:
959:
955:
950:
946:
926:
917:
916:
891:
887:
883:
878:
874:
871:
866:
863:
858:
850:
847:
844:
841:
838:
835:
832:
822:
811:
803:
799:
795:
792:
787:
783:
779:
776:
769:
765:
761:
758:
753:
749:
745:
737:
734:
731:
727:
723:
720:
710:
694:
689:
685:
681:
678:
675:
672:
667:
664:
661:
657:
653:
648:
644:
640:
637:
634:
631:
621:
602:
598:
593:
572:
542:
536:
525:
515:
499:
496:
493:
473:
470:
467:
452:
441:weight lattice
434:
424:
410:
403:positive roots
396:
377:
374:
328:
322:
302:
296:
250:
235:
234:
217:
216:
171:
169:
162:
155:
154:
69:
67:
60:
55:
29:
28:
26:
19:
9:
6:
4:
3:
2:
4800:
4789:
4786:
4784:
4781:
4779:
4776:
4775:
4773:
4763:
4759:
4756:
4752:
4751:
4742:
4738:
4734:
4728:
4724:
4719:
4716:
4712:
4707:
4702:
4699:: Art. B45a,
4698:
4694:
4693:
4687:
4684:
4680:
4678:
4675:
4672:
4671:0-8218-0770-6
4668:
4664:
4660:
4658:
4655:
4652:
4651:0-19-853489-2
4648:
4644:
4640:
4637:
4633:
4629:
4625:
4621:
4617:
4613:
4609:
4604:
4600:
4595:
4591:
4587:
4586:
4581:
4576:
4573:
4569:
4565:
4561:
4556:
4551:
4547:
4543:
4539:
4534:
4532:
4529:
4525:
4524:
4520:Haiman, Mark
4519:
4516:
4515:
4510:
4507:
4503:
4498:
4493:
4488:
4483:
4479:
4475:
4471:
4467:
4463:
4459:
4454:
4451:
4447:
4443:
4439:
4435:
4431:
4427:
4423:
4422:
4416:
4415:
4402:
4397:
4390:
4384:
4380:
4374:
4368:
4364:
4358:
4351:
4347:
4343:
4339:
4334:
4329:
4324:
4319:
4315:
4311:
4310:
4302:
4298:
4290:
4276:
4256:
4230:
4214:
4210:
4204:
4200:
4196:
4190:
4187:
4184:
4181:
4176:
4172:
4168:
4165:
4162:
4159:
4156:
4151:
4147:
4138:
4134:
4128:
4124:
4120:
4114:
4111:
4108:
4105:
4090:
4086:
4078:
4077:
4076:
4037:
4017:
4014:
4011:
3985:
3969:
3965:
3961:
3955:
3952:
3949:
3946:
3931:
3927:
3919:
3918:
3917:
3914:
3910:
3900:
3897:
3894:
3885:
3883:
3878:
3876:
3857:
3851:
3848:
3845:
3839:
3834:
3830:
3824:
3820:
3816:
3810:
3804:
3797:
3796:
3795:
3777:
3771:
3768:
3764:
3760:
3757:
3754:
3746:
3743:
3739:
3735:
3732:
3728:
3722:
3716:
3712:
3703:
3697:
3694:
3690:
3686:
3680:
3677:
3674:
3671:
3668:
3660:
3653:
3650:
3639:
3638:
3637:
3636:'s. We have
3621:
3617:
3593:
3590:
3587:
3584:
3581:
3573:
3566:
3563:
3550:
3549:
3532:
3524:
3520:
3501:
3495:
3472:
3466:
3446:
3423:
3417:
3391:
3388:
3385:
3382:
3379:
3371:
3367:
3363:
3352:
3346:
3343:
3340:
3336:
3327:
3321:
3317:
3313:
3310:
3299:
3293:
3290:
3287:
3283:
3279:
3273:
3270:
3267:
3264:
3261:
3253:
3249:
3241:
3240:
3239:
3222:
3219:
3216:
3213:
3210:
3202:
3198:
3174:
3171:
3168:
3165:
3162:
3154:
3150:
3141:
3140:integral form
3123:
3119:
3095:
3092:
3089:
3086:
3083:
3075:
3068:
3065:
3048:
3044:
3042:
3038:
3019:
3012:
3008:
3002:
2998:
2990:
2986:
2980:
2976:
2967:
2963:
2959:
2955:
2951:
2947:
2946:Young diagram
2926:
2922:
2913:
2907:
2904:
2901:
2897:
2888:
2882:
2879:
2876:
2872:
2864:
2851:
2848:
2845:
2841:
2837:
2831:
2828:
2825:
2822:
2819:
2811:
2804:
2801:
2790:
2789:
2788:
2772:
2768:
2745:
2738:
2735:
2724:
2719:
2705:
2699:
2696:
2693:
2685:
2682:
2678:
2662:
2660:
2655:
2650:
2646:
2642:
2638:
2634:
2630:
2626:
2622:
2617:
2600:
2593:
2577:
2573:
2569:
2564:
2560:
2552:
2536:
2532:
2528:
2523:
2519:
2511:
2495:
2491:
2487:
2482:
2478:
2470:
2454:
2450:
2446:
2441:
2437:
2429:
2413:
2409:
2403:
2399:
2395:
2390:
2386:
2380:
2376:
2372:
2367:
2363:
2357:
2353:
2349:
2344:
2340:
2334:
2330:
2326:
2321:
2317:
2311:
2307:
2303:
2298:
2294:
2288:
2284:
2280:
2275:
2263:
2262:
2261:
2256:
2251:
2247:
2242:
2238:
2234:
2230:
2225:
2221:
2216:
2212:
2208:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2154:
2150:
2144:
2140:
2132:
2128:
2122:
2118:
2108:
2103:
2091:
2090:
2089:
2070:
2056:
2048:
2043:
2032:
2029:
2024:
2020:
2012:
2011:
2010:
2008:
2004:
2000:
1996:
1994:
1986:n! conjecture
1983:
1981:
1976:
1974:
1972:
1967:
1963:
1959:
1955:
1951:
1947:
1943:
1936:
1918:
1914:
1906:
1887:
1884:
1881:
1865:
1847:
1840:
1837:
1825:
1821:
1817:
1811:
1807:
1797:
1795:
1791:
1787:
1768:
1757:
1753:
1744:
1740:
1731:
1728:
1721:
1717:
1712:
1704:
1700:
1695:
1691:
1688:
1680:
1676:
1669:
1658:
1654:
1645:
1641:
1632:
1629:
1622:
1618:
1613:
1605:
1601:
1596:
1592:
1589:
1581:
1577:
1566:
1565:
1564:
1562:
1559:at the point
1555:
1549:
1546:
1542:
1538:
1534:
1515:
1507:
1504:
1495:
1492:
1489:
1482:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1453:
1449:
1446:
1439:
1436:
1427:
1424:
1421:
1414:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1385:
1381:
1378:
1370:
1367:
1364:
1361:
1358:
1354:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1326:
1322:
1314:
1310:
1306:
1301:
1297:
1286:
1285:
1284:
1282:
1278:
1274:
1261:
1256:
1252:
1245:
1238:
1234:
1231:
1224:
1212:
1208:
1205:
1198:
1194:
1191:
1187:
1183:
1179:
1175:
1171:
1167:
1164:
1160:
1156:
1152:
1149:
1145:
1141:
1137:
1133:
1129:
1126:
1122:
1118:
1114:
1110:
1109:
1103:
1101:
1097:
1093:
1089:
1085:
1080:
1075:
1072:
1065:
1058:
1054:
1050:
1039:
1032:
1029:
1023:
1019:
1015:
1008:
990:
986:
980:
977:
973:
967:
964:
961:
957:
953:
948:
944:
936:
935:
934:
932:
925:
922:
914:
910:
906:
885:
876:
861:
856:
845:
839:
836:
833:
823:
809:
793:
790:
785:
781:
777:
759:
756:
751:
747:
735:
732:
729:
725:
721:
711:
708:
687:
683:
679:
676:
673:
665:
662:
659:
655:
651:
638:
635:
632:
622:
619:
615:
611:
606:
597:
594:
591:
586:
581:
577:
573:
570:
566:
562:
558:
557:group algebra
554:
550:
546:
543:
540:
537:
534:
530:
526:
523:
519:
516:
513:
497:
494:
491:
471:
468:
465:
457:
453:
450:
446:
442:
438:
435:
432:
428:
425:
422:
418:
414:
411:
408:
404:
400:
397:
394:
390:
386:
383:
382:
381:
373:
371:
367:
363:
359:
355:
347:
343:
339:
335:
331:
321:
317:
313:
309:
305:
295:
291:
287:
283:
279:
275:
271:
268:
264:
260:
256:
249:
246:
242:
231:
228:
213:
210:
202:
192:
188:
182:
181:
175:
170:
161:
160:
151:
148:
140:
129:
126:
122:
119:
115:
112:
108:
105:
101:
98: â
97:
93:
92:Find sources:
86:
80:
79:
75:
70:This article
68:
64:
59:
58:
53:
51:
44:
43:
38:
37:
32:
27:
18:
17:
4722:
4696:
4690:
4682:
4662:
4642:
4611:
4607:
4589:
4583:
4545:
4541:
4537:
4522:
4513:
4511:Mark Haiman
4469:
4465:
4461:
4425:
4419:
4412:Bibliography
4389:
4378:
4373:
4362:
4357:
4323:math/0409538
4313:
4307:
4301:
4248:
4003:
3906:
3898:
3895:
3891:
3879:
3872:
3793:
3551:
3548:polynomials.
3546:
3522:
3518:
3409:
3139:
3053:
3040:
3033:
2965:
2961:
2957:
2953:
2949:
2948:of shape Ό,
2943:
2722:
2720:
2668:
2656:
2648:
2644:
2640:
2628:
2620:
2618:
2615:
2254:
2249:
2245:
2240:
2236:
2232:
2228:
2223:
2219:
2214:
2210:
2206:
2204:
2087:
2006:
1992:
1989:
1979:
1977:
1970:
1961:
1957:
1953:
1949:
1945:
1941:
1934:
1823:
1819:
1815:
1809:
1805:
1803:
1783:
1560:
1553:
1550:
1544:
1540:
1530:
1280:
1276:
1272:
1270:
1254:
1250:
1243:
1236:
1222:
1210:
1196:
1185:
1177:
1173:
1169:
1162:
1158:
1154:
1147:
1139:
1131:
1124:
1116:
1112:
1099:
1095:
1087:
1083:
1078:
1076:
1070:
1063:
1056:
1052:
1048:
1046:
1037:
1030:
1025:λλ
1021:
1017:
1013:
1010:λμ
1006:
930:
923:
920:
918:
912:
908:
904:
617:
613:
609:
604:
595:
589:
584:
579:
575:
568:
564:
560:
552:
548:
544:
538:
532:
521:
517:
512:simple roots
448:
447:(containing
444:
436:
430:
426:
420:
412:
407:Weyl chamber
398:
392:
387:is a finite
384:
379:
333:
326:
319:
315:
311:
307:
300:
293:
289:
285:
277:
262:
258:
254:
247:
244:
238:
223:
205:
196:
177:
143:
134:
124:
117:
110:
103:
91:
71:
47:
40:
34:
33:Please help
30:
3913:L. Williams
3517:denote the
2964:. The term
2787:, they are
2003:Mark Haiman
1966:Mark Haiman
1090:) known as
529:Weyl vector
389:root system
370:conjectures
241:mathematics
191:introducing
4772:Categories
4401:1811.01024
4293:References
3909:S. Corteel
2637:Gorenstein
2635:(and even
2209:!, where (
2009:the space
1168:If we put
1053:orthogonal
417:Weyl group
376:Definition
350:1-variable
314:variables
267:orthogonal
199:April 2014
174:references
137:April 2014
107:newspapers
74:references
36:improve it
4628:0036-1410
4442:0003-486X
4383:MR1354144
4367:eudml.org
4342:0894-0347
4277:λ
4257:μ
4211:∑
4205:μ
4201:∑
4139:μ
4129:μ
4125:∑
4091:λ
4038:λ
4015:×
3966:∑
3932:λ
3907:In 2018,
3849:−
3840:⋅
3831:μ
3821:∑
3811:μ
3769:−
3744:−
3736:−
3717:μ
3704:μ
3695:−
3661:μ
3654:~
3622:μ
3574:μ
3567:~
3447:λ
3424:λ
3372:λ
3364:⋅
3314:−
3300:λ
3291:∈
3284:∏
3254:λ
3203:λ
3155:λ
3124:λ
3076:μ
3069:~
3020:⋯
3009:σ
2987:σ
2927:σ
2914:σ
2889:σ
2855:→
2852:μ
2846:σ
2842:∑
2812:μ
2805:~
2773:λ
2746:μ
2739:~
2686:μ
2683:λ
2570:−
2529:−
2488:−
2447:−
2396:−
2373:−
2350:−
2276:μ
2272:Δ
2185:≤
2170:≤
2104:μ
2100:Δ
2071:μ
2067:Δ
2053:∂
2040:∂
2025:μ
1919:λ
1848:μ
1841:~
1792:), using
1786:Cherednik
1758:ρ
1745:μ
1732:…
1718:ρ
1701:λ
1689:…
1681:μ
1659:ρ
1646:λ
1633:…
1619:ρ
1602:μ
1590:…
1582:λ
1505:−
1496:α
1490:α
1479:α
1470:ρ
1461:λ
1450:−
1428:α
1422:α
1411:α
1402:ρ
1393:λ
1382:−
1355:∏
1343:α
1334:∈
1331:α
1327:∏
1320:⟩
1315:λ
1302:λ
1294:⟨
991:μ
981:μ
978:λ
968:λ
965:≤
962:μ
958:∑
949:λ
870:Δ
865:¯
843:⟩
831:⟨
802:∞
786:α
768:∞
752:α
733:∈
730:α
726:∏
719:Δ
677:−
663:≥
656:∏
647:∞
555:) is the
527:Ï is the
498:μ
495:−
492:λ
472:λ
469:≤
466:μ
332:), where
274:Macdonald
42:talk page
4760:Some of
4506:15753285
1812:−1
1176:and let
1106:Examples
929:for λ â
567:for λ â
368:several
4741:1976581
4715:1817334
4677:1488699
4657:1354144
4636:0674768
4572:9253880
4531:2059359
4474:Bibcode
4450:2118632
4350:2138143
3794:where
1788: (
582:, then
439:is the
415:is the
187:improve
121:scholar
4739:
4729:
4713:
4669:
4649:
4634:
4626:
4570:
4504:
4497:554818
4494:
4448:
4440:
4348:
4340:
3459:, and
3410:where
3032:where
1188:, and
1041:μ
1034:λ
1005:where
705:, the
588:means
176:, but
123:
116:
109:
102:
94:
4701:arXiv
4568:S2CID
4550:arXiv
4446:JSTOR
4396:arXiv
4318:arXiv
1020:with
366:prove
325:,...,
299:,...,
128:JSTOR
114:books
4727:ISBN
4667:ISBN
4647:ISBN
4624:ISSN
4502:PMID
4466:PNAS
4438:ISSN
4338:ISSN
4050:and
3521:and
3488:and
2952:and
2001:and
1990:The
1960:and
1822:and
1790:1995
1368:<
1362:<
1346:>
1036:and
1027:= 1;
1016:and
919:The
603:λ â
360:and
344:and
340:and
100:news
4616:doi
4594:doi
4560:doi
4540:",
4492:PMC
4482:doi
4470:102
4464:",
4430:doi
4426:141
4328:doi
3523:leg
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