Table Of ContentSymmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 034, 57 pages
Notes on Schubert, Grothendieck
and Key Polynomials
Anatol N. KIRILLOV †‡§
† Research Institute of Mathematical Sciences (RIMS), Kyoto, Sakyo-ku 606-8502, Japan
E-mail: [email protected]
URL: http://www.kurims.kyoto-u.ac.jp/~kirillov/
‡ The Kavli Institute for the Physics and Mathematics of the Universe (IPMU),
5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
6
1 § Department of Mathematics, National Research University Higher School of Economics,
0
2 7 Vavilova Str., 117312, Moscow, Russia
r Received March 26, 2015, in final form February 28, 2016; Published online March 29, 2016
p
A http://dx.doi.org/10.3842/SIGMA.2016.034
4
Abstract. We introduce common generalization of (double) Schubert, Grothendieck, De-
] mazure, dual and stable Grothendieck polynomials, and Di Francesco–Zinn-Justin polyno-
O
mials. Our approach is based on the study of algebraic and combinatorial properties of the
C reduced rectangular plactic algebra and associated Cauchy kernels.
.
h Key words: plactic monoid and reduced plactic algebras; nilCoxeter and idCoxeter alge-
t
a bras; Schubert, β-Grothendieck, key and (double) key-Grothendieck, and Di Francesco–
m Zinn-Justin polynomials; Cauchy’s type kernels and symmetric, totally symmetric plane
partitions, and alternating sign matrices; noncrossing Dyck paths and (rectangular) Schu-
[
bert polynomials; multi-parameter deformations of Genocchi numbers of the first and the
3
second types; Gandhi–Dumont polynomials and (staircase) Schubert polynomials; double
v
affine nilCoxeter algebras
7
3
2010 Mathematics Subject Classification: 05E05; 05E10; 05A19
3
7
0 To the memory of Alexander Grothendieck (1928–2014)
.
1
Contents
0
5
1 1 Introduction 2
:
v 2 Plactic, nilplactic and idplactic algebras 10
i
X
3 Divided difference operators 17
r
a
4 Schubert, Grothendieck and key polynomials 18
5 Cauchy kernel 28
5.1 Plactic algebra P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
n
5.2 Nilplactic algebra NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
n
5.3 Idplactic algebra IP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
n
5.4 NilCoxeter algebra NC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
n
5.5 IdCoxeter algebras IC± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
n
6 F-kernel and symmetric plane partitions 41
A Appendix 45
A.1 Some explicit formulas for n=4 and compositions α such that α ≤n−i for i=1,2,... 45
i
A.2 MacMeille completion of a partially ordered set . . . . . . . . . . . . . . . . . . . . . . . . 52
References 54
2 A.N. Kirillov
Extended abstract
We introduce certain finite-dimensional algebras denoted by PC and PF which are certain
n n,m
quotients of the plactic algebra P , which had been introduced by A. Lascoux and M.-P. Schu¨t-
n
zenberger [46]. We show that dim(PF ) is equal to the number of symmetric plane partitions
n,k
fitting inside the box n×k×k, dim(PC ) is equal to the number of alternating sign matrices
n
of size n×n, moreover,
dim(PF ) = TSPP(n+1)×TSSCPP(n),
n,n
dim(PF ) = TSPP(n+1)×TSSCPP(n+1),
n,n+1
1
dim(PF ) = dim(PF ), dim(PF ) = dim(PF ),
n+2,n n,n+1 n+3,n n+1,n+1
2
and study decomposition of the Cauchy kernels corresponding to the algebras PC and PF ;
n n,m
as well as introduce polynomials which are common generalizations of the (double) Schubert,
β-Grothendieck, Demazure (known also as key polynomials), (plactic) key-Grothendieck, (plac-
tic) Stanley and stable β-Grothendieck polynomials. Using a family of the Hecke type divided
difference operators we introduce polynomials which are common generalizations of the Schu-
bert,β-Grothendieck,dualβ-Grothendieck,β-Demazure–Grothendieck,andDiFrancesco–Zinn-
Justinpolynomials. WealsointroduceandstudysomepropertiesofthedoubleaffinenilCoxeter
algebras and related polynomials, put forward a q-deformed version of the Knuth relations and
plactic algebra.
1 Introduction
The Grothendieck polynomials had been introduced by A. Lascoux and M.-P. Schu¨tzenberger
in [48] and studied in detail in [39]. There are two equivalent versions of the Grothendieck
polynomials depending on a choice of a basis in the Grothendieck ring K(cid:63)(Fl ) of the complete
n
flag variety Fl . The basis {exp(ξ ),...,exp(ξ )} in K∗(Fl ) is one choice, and another choice
n 1 n n
is the basis {1 − exp(−ξ ), 1 ≤ j ≤ n}, where {ξ , 1 ≤ j ≤ n} denote the Chern classes of
j j
the tautological linear bundles L over the flag variety Fl . In the present paper we use the
j n
basis in a deformed Grothendieck ring K∗,β(F ) of the flag variety Fl generated by the set of
n n
(β)
elements {x = x = 1−exp(βξ ), i = 1,...,n}. This basis has been introduced and used for
i i i
construction of the β-Grothendieck polynomials in [17, 19].
A basis in the classical Grothendieck ring of the flag variety in question corresponds to
(cid:8) (β) (cid:9)
the choice β = −1. For arbitrary β the ring generated by the elements x , 1 ≤ i ≤ n
i
has been identified with the Grothendieck ring corresponding to the generalized cohomology
theory associated with the multiplicative formal group law F(x,y) = x + y + βxy, see [23].
The Grothendieck polynomials corresponding to the classical K-theory ring K(cid:63)(Fl ), i.e., the
n
case β = −1, had been studied in depth by A. Lascoux and M.-P. Schu¨tzenberger in [49]. The
β-Grothendieck polynomials has been studied in [17, 18, 23].
The plactic monoid over a finite totally ordered set A = {a < b < c < ··· < d} is the
quotient of the free monoid generated by elements from A subject to the elementary Knuth
transformations [29]
bca = bac & acb = cab, and bab = bba & aba = baa, (1.1)
for any triple {a < b < c} ⊂ A.
To our knowledge, the concept of “plactic monoid” has its origins in a paper by C. Schen-
sted [64], concerning the study of the longest increasing subsequence of a permutation, and
Notes on Schubert, Grothendieck and Key Polynomials 3
a paper by D. Knuth [29], concerning the study of combinatorial and algebraic properties of the
Robinson–Schensted correspondence1.
Asfarasweknow, thismonoidandthe(unital)algebraP(A)correspondingtothatmonoid2,
had been introduced, studied and used by M.-P. Schu¨tzenberger, see [65, Section 5], to give
the first complete proof of the famous Littlewood–Richardson rule in the theory of symmetric
functions. A bit later this monoid, was named the “mono¨ıde plaxique” and studied in depth by
A. Lascoux and M.-P. Schu¨tzenberger [46]. The algebra corresponding to the plactic monoid is
commonlyknownasplactic algebra. Oneofthebasicpropertiesoftheplacticalgebra[65]isthat
it contains the distinguished commutative subalgebra which is generated by noncommutative
elementary (quasi-symmetric) polynomials3
(cid:88)
e (A ) = a a ···a , k = 1,...,n,
k n i1 i2 ik
i1>i2>···>ik
see, e.g., [65, Corollary 5.9] and [16].
We refer the reader to nice written overview [44] of the basic properties and applications of
the plactic monoid in combinatorics.
It is easy to see that the plactic relations for two letters a < b, namely,
aba = baa, bab = bba,
imply the commutativity of noncommutative elementary polynomials in two variables. In other
words, the plactic relations for two letters imply that
ba(a+b) = (a+b)ba, a < b.
It has been proved in [16] that these relations together with the Knuth relations (1.1) for three
letters a < b < c, imply the commutativity of noncommutative elementary quasi-symmetric
polynomials for any number of variables.
In the present paper we prove that in fact the commutativity of noncommutative elementary
quasi-symmetricpolynomialsforn = 2andn = 3impliesthecommutativityofthatpolynomials
for all n, see Theorem 2.234.
One of the main objectives of the present paper is to study combinatorial properties of the
generalized plactic Cauchy kernel
n−1 i
(cid:89) (cid:89)
C(P ,U) = (1+p u ) ,
n i,j−i+1 j
i=1 j=n−1
where P stands for the set of parameters {p , 2 ≤ i+j ≤ n+1, i > 1, j > 1}, and U := U
n ij n
stands for a certain noncommutative algebra we are interested in, see Section 5.
We also want to bring to the attention of the reader on some interesting combinatorial
properties of rectangular Cauchy kernels
n−1 1
(cid:89) (cid:89)
F(P ,U) = (1+p u ) ,
n,m i,i−j+1(m) j
i=1 j=m−1
where P = {p } ; see Definition 6.6 for the meaning of symbol a(m).
n,m ij 1≤i≤n
1≤j≤m
1See, e.g., https://en.wikipedia.org/wiki/Robinson-Schensted_correspondence.
2If A = {1 < 2 < ··· < n}, the elements of the algebra P(A) can be identified with semistandard Young
tableaux. ItwasdiscoveredbyD.Knuth[29]thatmoduloKnuth equivalencetheequivalenceclassesofsemistan-
dardYoungtableauxformanalgebra,andhehasnamedthisalgebrabytableauxalgebra. Itiseasilyseenthatthe
tableaux algebra introduced by D. Knuth is isomorphic to the algebra introduced by M.-P. Schu¨tzenberger [65].
3See, e.g., [36] for definition of noncommutative quasi-symmetric functions and polynomials.
4Letusstressthatconditionsnecessaryandsufficienttoassurethecommutativityofnoncommutativeelemen-
tarypolynomialsforthenumberofvariablesequalsn=2andn=3turnouttobeweakerthenthatlistedin[16].
4 A.N. Kirillov
Wetreatthesekernelsinthe(reduced)placticalgebrasPC andPF correspondingly. The
n n,m
algebras PC and PF are finite-dimensional and have bases parameterized by certain Young
n n,m
tableaux described in Sections 5.1 and 6 correspondingly. Decomposition of the rectangular
Cauchy kernel with respect to the basis in the algebra PF mentioned above, gives rise
n,m
to a set of polynomials which are common generalizations of the (double) Schubert. β-Gro-
thendieck, DemazureandStanleypolynomials. Tobemoreprecise, thepolynomialslistedabove
correspond to certain quotients of the plactic algebra PF and appropriate specializations of
n,m
parameters {p } involved in our definition of polynomials U ({p }), see Section 6.
ij α ij
As it was pointed out in the beginning of Introduction, the Knuth (or plactic) relations (1.1)
have been discovered in [29] in the course of the study of algebraic and combinatorial proper-
ties of the Robinson–Schensted correspondence. Motivated by the study of basic properties of
a quantum version of the tropical/geometric Robinson–Schensted–Knuth correspondence – work
inprogress,butsee[3,26,28,59,60]fordefinitionandbasicpropertiesofthetropical/geometric
RSK, – the author of the present paper came to a discovery that certain deformations of the
Knuth relations preserve the Hilbert series (resp. the Hilbert polynomials) of the plactic alge-
bras P and F (resp. the algebras PC and PF ).
n n n n
More precisely, let {q ,...,q } be a set of (mutually commuting) parameters, and U :=
2 n n
{u ,...,u } be a set of generators of the free associative algebra over Q of rank n. Let Y,Z ⊂
1 n
[1,n] be subsets such that Y ∪Z = [1,n] and Y ∩Z = ∅. Let us set p(a) = 0, if a ∈ Y and
p(a) = 1, if a ∈ Z. Define q-deformed super Knuth relations among the generators u ,...,u
1 n
as follows:
SPL : (−1)p(i)p(k)q u u u = u u u , i < j ≤ k,
q k j i k j k i
(−1)p(i)p(k)q u u u = u u u , i ≤ j < k.
k i k j k i j
We define
• q-deformed superplactic algebra SQP to be the quotient of the free associative alge-
n
bra Q(cid:104)u ,...,u (cid:105) by the two-sided ideal generated by the set of q-deformed Knuth rela-
1 n
tions (SPL ),
q
• reduced q-deformed superplactic algebras SQPC and SQPF to be the quotient of the
n n,m
algebra SQP by the two-sided ideals described in Definitions 5.18 and 6.7 correspon-
n
dingly.
We state
Conjecture 1.1. The algebra SQP and the algebras SQPC and SQPF , are flat defor-
n n n,m
mations of the algebras P , PC and PF correspondingly.
n n n,m
In fact one can consider more general deformation of the Knuth relations, for example take
a set of parameters Q := {q , 1 ≤ i < k ≤ n} and impose on the set of generators {u ,...,u }
ik 1 n
the following relations
q u u u = u u u , i < j ≤ k, q u u u = u u u , i ≤ j < k.
ik j i k j k i ik i k j k i j
However we don’t know how to describe a set of conditions on parameters Q which imply the
flatnessofthecorrespondingquotientalgebra(s), aswellaswedon’tknowaninterpretationand
dimension of the algebras SQPC and SQPF for a “generic” values of parameters Q. We
n n,m
expectthedimensionofalgebrasSQPC andSQPF eachdependspiece-wisepolynomiallyon
n n,m
asetofparameters{q ∈ Z , 1 ≤ i < j ≤ n}, andposeaproblemtodescribeitspolynomiality
ij ≥0
chambers.
We also mention and leave for a separate publication(s), the case of algebras and polynomials
associated with superplactic monoid [38, 56], which corresponds to the relations SPL with
q
Notes on Schubert, Grothendieck and Key Polynomials 5
q = 1, ∀i. Finally we point out an interesting and important paper [55] wherein the case
i
Z = ∅, and the all deformation parameters are equal to each other, has been independently
introduced and studied in depth.
Let us repeat that the important property of plactic algebras P is that the noncommutative
n
elementary polynomials
(cid:88)
e (u ,...,n ) := u ···u , k = 1,...,n−1,
k 1 n−1 a1 ak
n−1≥a1≥a2≥ak≥1
generate a commutative subalgebra inside of the plactic algebra P , see, e.g., [16, 46]. Therefore
n
all our finite-dimensional algebras introduced in the present paper, have a distinguished finite-
dimensional commutative subalgebra. We have in mined to describe these algebras explicitly in
a separate publication.
In Section 2 we state and prove necessary and sufficient conditions in order the elementary
noncommutative polynomials form a mutually commuting family. Surprisingly enough to check
the commutativity of noncommutative elementary polynomials for any n, it’s enough to check
these conditions only for n = 2,3. However a combinatorial meaning of a generalization of the
Lascoux–Schu¨tzenberger plactic algebra P obtained in this way, is still missing.
n
TheplacticalgebraPF introducedinSection6,hasamonomialbasisparametrizedbythe
n,m
set of Young tableaux of shape λ ⊂ (nm) filled by the numbers from the set {1,...,m}. In the
case n = m it is well-known [20, 35, 58], that this number is equal to the number of symmetric
plane partitions fitting inside the cube n × n × n. Surprisingly enough this number admits
a factorization in the product of the number of totally symmetric plane partitions (TSPP) by
the number of totally symmetric self-complementary plane partitions (TSSCPP) fit inside the
same cube. A similar phenomenon happens if |m−n| ≤ 2, see Section 6. More precisely, we
add to the well-known equalities
(cid:18) (cid:19)
2n+1
#|B | = 2n, #|B | = , #|B | = 2nCat [67,A003645],
1,n 2,n 3,n n+1
n
1
#|B | = Cat Cat [67,A000356],
4,n n+1 n+2
2
(cid:0)n+5(cid:1)(cid:0)n+7(cid:1)(cid:0)n+9(cid:1)
#|B | = 5 7 9 [67,A133348],
n,5 (cid:0)n+2(cid:1)(cid:0)n+4(cid:1)
2 4
the following relations
#|B | = TSPP(n+1)×ASM(n), #|B | = TSPP(n+1)×ASM(n+1),
n,n n,n+1
1
#|B | = #|B |, #|B | = #|B |,
n+2,n n,n+1 n+3,n n+1,n+1
2
#|PP(n)| = #|TSSCPP(n)|×#|ASMHT(2n)| = #|CSSCPP(2n)|×#|CSPP(n)|,
#|CSPP(2n)| = #|TSPP(2n)|×#|CSTCPP(2n)|,
#|CSPP(2n+1)| = 22n#|TSPP(2n+1)|×#|TSPP(2n)|,
where PP(n) stands for the set of plane partitions fit in a cub of size n×n×n; AMSHT(2n)
denotes the set of alternating sign matrices of size 2n × 2n invariant under a half-turn and
CSSPP(2n) denotes the set of cyclically symmetric self-complementary plane partitions fitting
inside a cub of size 2n × 2n × 2n, see, e.g., [6]; CSTCPP(n) stands for the set of cyclically
symmetric transpose complementary plane partitions fitting inside a cub of size 2n×2n×2n,
see, e.g., [67, A051255]. See Section 6 for the definition of the sets B and examples. In
n,m
Exercise 6.3 we state some (new) divisibility properties of the numbers #|B |.
n+4,n
6 A.N. Kirillov
Itiswell-knownthatASMHT(2n) = ASM(n)×CSPP(n),whereCSPP(n)denotesthenumber
of cyclically symmetric plane partitions fitting inside n-cube, and CSSCPP(2n) = ASM(n)2,
see, e.g., [6, 37] and [67, A006366].
Problem 1.2.
• Construct bijection between the set of plane partitions fit inside n-cube and the set of
(ordered) triples (π ,π ,℘), where (π ,π ) is a pair of TSSCPP(n) and ℘ is a cyclically
1 2 1 2
symmetric plane partition fitting inside n-cube.
• Describe the involution κ: PP(n) −→ PP(n) which is induced by the involution (π ,π ,℘)
1 2
−→ (π ,π ,℘) on the set TSSCPP(n)×TSSCP(n)×CSPP(n), and its fixed points. Clearly
2 1
one has #|Fix(κ)| = ASMHT(2n).
• Characterize pairs of plane partitions (Π ,Π ) ∈ PP(n)×PP(n) such that
1 2
(a) ℘(Π ) = ℘(Π ); (b) (π (Π ),π (Π )) = (π (Π ),π (Π )).
1 2 1 1 2 1 1 2 2 2
These relations have straightforward proofs based on the explicit product formulas for the
numbers
(cid:89) n+i+j +k−1
#|SPP(n)| = ,
i+j +k−1
1≤i≤j≤k
n n n
(cid:89)(cid:89)(cid:89) i+j +k−1
#|TSPP(n)| = ,
i+j +k−2
i=1j=ik=j
n n−1 n (cid:0)2n+i(cid:1)
(cid:89)(cid:89) 3n−i−j (cid:89)
#|PP(n)| = = n ,
2n−i−j (cid:0)n+i(cid:1)
i=0j=1 i=1 n
but bijective proofs of these identities are an open problem.
It follows from [42, 46] that the dimension of the (reduced) plactic algebra PC is equal
n
to the number of alternating sign matrices of size n×n (note that ASM(n) = TSSCPP(n)).
Thereforethekey-GrothendieckpolynomialscanbeobtainedfromU-polynomials(seeSection6,
Theorem 6.12) after the specialization p = 0, if i+j > n+1.
ij
In Section 4 following [27] we introduce and study a family of polynomials which are a com-
mon generalization of the Schubert, β-Grothendieck, dual β-Grothendieck, β-Demazure, β-key-
Grothendieck, Bott–Samelson and q-Demazure polynomials, Whittaker functions (see [7] and
Lemma 4.18) and Di Francesco–Zinn-Justin polynomials (see Section 4). Namely, for any per-
mutation w ∈ S and composition ζ ⊂ δ := (n−1,n−2,...,2,1), we introduce polynomials
n n
KN(β,α,γ,h)(X ) = h(cid:96)(w)T ···T (cid:0)xδn(cid:1),
w n si1 si(cid:96)
KD(β,α,γ,h)(X ) = h(cid:96)(vζ)T ···T (cid:0)xζ+(cid:1),
ζ n si1 si(cid:96)
where
(β,α,γ,h)
T := T = −α+((α+β +γ)x +γx +h
i i i i+1
+h−1(α+γ)(β+γ)x x )∂ , i = 1,...,n−1,
i i+1 i,i+1
denote a collection of divided difference operators which satisfy the Coxeter and Hecke relations
T T T = T T T , if |i−j| = 1; T T = T T , if |i−j| ≥ 2,
i j i j i j i j j i
T2 = (β −α)T +βα, i = 1,...,n−1;
i i
Notes on Schubert, Grothendieck and Key Polynomials 7
by definition for any permutation w ∈ S we set
n
T := T ···T ,
w si1 si(cid:96)
for any reduced decomposition w = s ···s of a permutation in question; ζ+ denotes a unique
i1 i(cid:96)
partition obtained from ζ by ordering its parts, and v ∈ S denotes the minimal length permu-
ζ n
tation such that v (ζ) = ζ+.
ζ
Assume that h = 1.5 If α = γ = 0, these polynomials coincide with the β-Grothendieck
polynomials [17], if β = α = 1, γ = 0 these polynomials coincide with the Di Francesco–Zinn-
Justin polynomials [12], if β = γ = 0, these polynomials coincide with dual α-Grothendieck
polynomials H(α)(X ),6 where by definition we set X := (x ,...,x ).
w n n 1 n
Conjecture 1.3. For any permutation w ∈ S and any composition ζ ⊂ δ , polynomials
n n
(β,α,γ,h) (α,β,γ,h)
KN (X ) and KD (X ) have nonnegative coefficients, i.e.,
w n ζ n
KN(β,α,γ,h)(X ) ∈ N[α,β,γ,h][X ], KD(β,α,γ,h)(X ) ∈ N[α,β,γ,h][X ].
w n n ζ n n
We expect that these polynomials have some geometrical meaning to be discovered.
More generally we study divided difference type operators of the form
(a,b,c,h,e)
T := T = a+(bx +cx +h+ex x )∂ ,
ij ij i j i j ij
depending on parameters a, b, c, h, e and satisfying the 2D-Coxeter relations
T T T = T T T , 1 ≤ i < j < k ≤ n,
ij jk ij jk ij jk
T T = T T , if {i,j}∩{k,l} = ∅.
ij kl kl ij
We find that the necessary and sufficient condition which ensure the validity of the 2D-Coxeter
relations is the following relation among the parameters7:
(a+b)(a−c)+he = 0.
Therefore, if the above relation between parameters a, b, c, h, e holds, then for any permu-
tation w ∈ S the operator
n
T := T(a,b,c,h,e) = T(a,b,c,h,e)···T(a,b,c,h,e),
w w i1 i(cid:96)
5Clearly that if h (cid:54)= 0, then after rescaling parameters α, β and γ one can assume that h = 1. However,
see, e.g., [27, Section 5], the parameter h plays important role in the study of different specializations of the
variables x , 1≤i≤n−1 and parameters α, β and γ.
i
6Toavoidthereader’sconfusion,letusexplainthatinourpaperweusetheletterαeitherasthelowerindexto
denoteacomposition,orastheupperindextodenoteaparameterwhichappearsincertainpolynomialstreatedin
ourpaper. ForexampleH(α)(X )denotesthedualα-Grothendieckpolynomialcorrespondingtoacompositionα.
α n
Notethattheα-GrothendieckpolynomialG(α)(X )canbeobtainedfromthepolynomialG(β)(X )byreplacingβ
w n w n
by α.
7In other words, the divided difference operators {T := T(a,b,c,h,e)} which obey the 2D-Coxeter relations,
ij ij
have the following form:
(cid:18) (a+b) (cid:19)(cid:16) c−a (cid:17)
T(a,b,c,h) = 1+ x 1+ x ∂ +aσ , if h(cid:54)=0, a, b, c arbitrary.
ij h i h j ij ij
If h=0, then either T =((c−a)x +ex x )∂ +aσ , or T =(a+b)x +ex x )∂ +aσ , where σ stands
ij j i j ij ij ij i i j ij ij ij
for the exchange operator: σ (F(z ,z ))=F(z ,z ).
ij i j j i
8 A.N. Kirillov
where w = s ···s is any reduced decomposition of w, is well-defined. Hence under the
i1 i(cid:96)
same assumption on parameters, for any permutation w ∈ S one can attach the well-defined
n
polynomial
(cid:18) (cid:19)
G(a,b,c,h,e)(X,Y) := T(x)(a,b,c,h,e) (cid:89) (x +y ) ,
w w i j
i≥1,j≥1
i+j≤n+1
and in much the same fashion to define polynomials
D(a,b,c,h,e)(X,Y) := T(x)(a,b,c,h,e)(cid:0)xα+(cid:1)
α wα
for any composition α such that α ≤ n−i, ∀i. We have used the notation T(x)(a,b,c,h,e) to point
i w
out that this operator acts only on the variables X = (x ,...,x ); for any composition α ∈ Zn ,
1 n ≥0
α+ denotes a unique partition obtained from α by reordering its parts in (weakly) decreasing
order, and w denotes a unique minimal length permutation in the symmetric group S such
α n
that w (α) = α+.
α
In the present paper we are interested in to list a conditions on parameters A := {a,b,c,h,e}
with the constraint
(a+b)(a−c)+he = 0,
(a,b,c,h,e) (a,b,c,h,e)
which ensure that the above polynomials G (X) and D (X) or their specialization
w α
x = 1, ∀i, have nonnegative coefficients. We state the following conjectures:
i
• KN(β,α,γ)(X ) ∈ N[α,β,γ][X ],
w n n
• G(−b,a+b+c,c,1,(b+c)(a+c)(X ) ∈ N[a,b,c][X ],
w n n
• G(−b,a+b+c,c+d,1,(b+c+d)(a+c)(x = 1, ∀i) ∈ N[a,b,c,d], where a, b, c, d are free parameters.
w i
In the present paper we treat the case
A = (−β,β +α+γ,γ,1,(α+γ)(β +γ)). (1.2)
As it was pointed above, in this case polynomials GA(X) are common generalization of Schu-
w
bert, β-Grothendieck and dual β-Grothendieck, and Di Francesco–Zinn-Justin polynomials. We
expect a certain interpretation of the polynomials GA for general β, α and γ.
w
As it was pointed out earlier, one of the basic properties of the plactic monoid P is that the
n
noncommutative elementary symmetric polynomials {e (u ,...,u )} generate a com-
k 1 n−1 1≤k≤n−1
mutative subalgebra in the plactic algebra in question. One can reformulate this statement as
follows. Consider the generating function
i i
(cid:89) (cid:88)
A (x) := (1+xu ) = e (u ,...,u )xi−a,
i a a n−1 i
a=n−1 a=0
where we set e (U) = 1. Then the commutativity property of noncommutative elementary
0
symmetric polynomials is equivalent to the following commutativity relation in the plactic as
well as in the generic plactic, algebras P and P [16], and Theorem 2.23,
n n
A (x)A (y) = A (y)A (x), 1 ≤ i ≤ n−1.
i i i i
Now let us consider the Cauchy kernel
C(P ,U) = A (z )···A (z ),
n 1 1 n−1 n−1
Notes on Schubert, Grothendieck and Key Polynomials 9
where we assume that the pairwise commuting variables z ,...,z commute with the all
1 n−1
generators of the algebras Pn and Pn. In what follows we consider the natural completion P(cid:98)n of
the plactic algebra P to allow consider elements of the form (1+xu )−1. Elements of this form
n i
exist in any Hecke type quotient of the plactic algebra P(cid:101)n. Having in mind this assumption,
let us compute the action of divided difference operators ∂z on the Cauchy kernel. In the
i,i+1
computation below, the commutativity property of the elements A (x) and A (y) plays the key
i i
role. Let us start computation of ∂z (C(P ,U)) = ∂z (A (z )···A (z )). First of all
i,i+1 n i,i+1 1 1 n−1 n−1
writeA (z ) = A (z )(1+z u )−1. AccordingtothebasicpropertyoftheelementsA (x),
i+1 i+1 i i+1 i+1 i i
one sees that the expression A (z )A (z ) is symmetric with respect to z and z , and hence
i i i i+1 i i+1
is invariant under the action of divided difference operator ∂z . Therefore,
i,i+1
∂z (C(P ,U)) = A (z )···A (z )A (z )∂z (cid:0)(1+z u )−1(cid:1)
i,i+1 n 1 1 i i i i+1 i,i+1 i+1 i
×A (z )···A (z ).
i+2 i+2 n−1 n−1
It is clearly seen that ∂z ((1+z u )−1) = (1+z u )−1(1+z u )−1u . Therefore,
i,i+1 i+1 i i i i+1 i i
∂z (C(P ,U)) = A (z )···A (z )A (z )(1+z u )−1u A (z )···A (z ).
i,i+1 n 1 1 i i i+1 i+1 i i i i+2 i+2 n−1 n−1
It is easy to see that if one adds Hecke’s type relations on the generators
u2 = (a+b)u +ab, i = 1,...,n−1,
i i
then
u −zab
(1+zu )−1u = i .
i i
(1+bz)(1−az)
Therefore in the quotient of the plactic algebra P by the Hecke type relations listed above and
n
by the “locality” relations
u u = u u , if |i−j| ≥ 2,
i j j i
one obtains
(cid:18) (cid:19)
e −b
(−b+(1+z b))∂z (A (z )···A (z )) = (A (z )···A (z )) i .
i i,i+1 1 1 n−1 n−1 1 1 n−1 n−1 1−az
i
Finally, if a = 0, then the above identity takes the following form
∂z ((1+z b)A (z )···A (z )) = (A (z )···A (z ))(e −b).
i,i+1 i+1 1 1 n−1 n−1 1 1 n−1 n−1 i
In other words the above identity is equivalent to the statement [19] that in the idCoxeter
algebra IC the Cauchy kernel C(P ,U) is the generating function for the b-Grothendieck poly-
n n
nomials. Moreover, each (generalized) double b-Grothendieck polynomial is a positive linear
combination of the key-Grothendieck polynomials.
Aproofofthisstatementisacorollaryofthemoregeneralstatementwhichwillbefrequently
used throughout the present paper, namely, if an equivalence relation ≈ is a refinement of
2
that ≈ , that is if assumption a≈ b =⇒ a≈ b holds ∀a, b, then each equivalence class w.r.t.
1 2 1
relation ≈ is disjoint union of the equivalence classes w.r.t. relation ≈ .
1 2
In the special case b = −1 and P = x +y if 2 ≤ i+j ≤ n+1, p = 0, if i+j > n+1,
ij i j ij
this result had been stated in [43].
As a possible mean to define affine versions of polynomials treated in the present paper, we
introducethedouble affine nilCoxeter algebra of type Aandgive constructionofa genericfamily
of Hecke’s type elements8 we will be put to use in the present paper.
8Remindthatbythenamea family of Hecke’s type elementswemeanasetofelements{e ,...,e }suchthat
1 n
e2 =Ae +B,A,B areparameters(Hecketyperelations),e e =e e ,if|i−j|≥2,e e e =e e e ,if|i−j|=1
i i i j j i i j i j i j
(Coxeter relations).
10 A.N. Kirillov
In Section 5.1 we suggest a common generalization of some combinatorial formulas from [10]
and [21]. Namely, we give explicit formula
(cid:89) N −i−j +1 (cid:89) N +i+j −1
(1.3)
i+j −1 i+j −1
1≤i≤k,1≤j≤n 1≤i≤k,1≤j≤n
j−i≤n−k j−i>n−k
for the number of k-tuples of noncrossing Dyck paths connecting the points (0,0) and (N,N −
n−k). Interpretations of the number (1.3) as the number of certain k-triangulations of a convex
(N +1)-gon, or that of certain alternating sign matrices of size N ×N, are interesting tasks.
In the case N = n + k we recover the [10, r.h.s. of formula (2)]. In the case k = 2 our
formula (1.3) is equivalent to that obtained in [21]. Our proof that the number (1.3) counts
certain k-tuples of noncrossing Dyck paths is based on the study of combinatorial properties of
the so-called column multi-Schur functions s∗(X ) introduced in Theorem 5.6, cf. [58, 72]. In
λ n
particularweshowthatforrectangularpartitionλ = (nk)thepolynomials∗(X )isessentially
λ n+k
coincide with the Schubert polynomial corresponding to the Richardson permutation 1k×w(n).
0
We introduce also a multivariable deformation of the numbers ASM(n), namely,
(cid:88)
ASM(X ;t) := s∗(X)t|λ|.
n−1 λ
λ⊂δn
Finally, in Section 5.1 we give combinatorial interpretations of rectangular and staircase compo-
nentsoftherefinedTSSCPvector[12]intermsofk-fansofnoncrossingDyckpathsinrectangular
case and Gandhi–Dumont polynomials and Genocchi numbers in staircase case.
As Appendix we include several examples of polynomials studied in the present paper to
illustrate results obtained in these notes. We also include an expository text concerning the
MacNeille completion of a poset to draw attention of the reader to this subject. It is an exami-
nationoftheMacNeillecompletionoftheposetassociatedwiththe(strong)Bruhatorderonthe
symmetric group, that was one of the main streams of the study in the present paper. Namely,
ourconcernwasthechallengehowtoattachtoeachedgeeoftheMacNeillecompletionMN(S )
n
of the Bruhat order poset on the symmetric group S an operator ∂ acting on the ring of
n e
polynomials Z[X ], such that ∂ (K ) = K together with compatibility conditions among
n e h(e) t(e)
the set of operators {∂ } , that is for any two vertices of MN(S ), say α and β, and
e e∈MN(Sn) n
a path p in the MacNeille completion which connects these vertices, the naturally defined
α,β
operator ∂ depends only on the vertices α and β taken, and doesn’t depend on a path p
pα,β α,β
selected. As far as I know, this problem is still open.
2 Plactic, nilplactic and idplactic algebras
Definition 2.1 ([46]). Theplactic algebra P isan(unital)associativealgebraoverZgenerated
n
by elements {u ,...,u } subject to the set of relations
1 n−1
(PL1) u u u = u u u , u u u = u u u , if i < j < k,
j i k j k i i k j k i j
(PL2) u u u = u u u , u u u = u u u , if i < j.
i j i j i i j i j j j i
Proposition 2.2 ([46]). Tableau words9 in the alphabet U = {u ,...,u } form a basis in the
1 n−1
plactic algebra P .
n
9For the reader convenience we recall a definition of a tableau word. Let T be a (regular shape) semistandard
Young tableau. The tableau word w(T) associated with T is the reading word of T is the sequence of entries
ofT obtainedbyconcatenatingthecolumnsofT bottomtotopconsecutivelystartingfromthefirstcolumn. For
