Doubly-refined enumeration of Alternating Sign Matrices and determinants of 2-staircase Schur functions

0.33 MB·English

by  Philippe Biane

1
1
Doubly-refined enumeration of Alternating Sign Matrices
0
2 and determinants of 2-staircase Schur functions

n
a Philippe Biane, Luigi Cantini, and Andrea Sportiello
J

8
1 Abstract. WeproveadeterminantalidentityconcerningSchurfunctionsfor
2-staircasediagramsλ=(ℓn+ℓ′,ℓn,ℓ(n−1)+ℓ′,ℓ(n−1),···,ℓ+ℓ′,ℓ,ℓ′,0).
] Whenℓ=1andℓ′=0thesefunctionsarerelatedtothepartitionfunctionof
O
the 6-vertex model at the combinatorial point and hence to enumerations of
C AlternatingSignMatrices. Aconsequenceofourresultisanidentityconcern-
ingthedoubly-refinedenumerations ofAlternatingSignMatrices.
.
h
t January 19, 2011
a
m
[

1
v 1. Introduction
7
1.1. Alternating Sign Matrices. An alternating sign matrix (ASM) is a
2
4 square matrix with entries in 1,0,+1 , such that on each line and on each
{− }
3 column,ifoneforgetsthe0’s,the+1’sand 1’salternate,andthesumofeachline
1. andeachcolumnis equalto 1. Itis a famou−s combinatorialresultthatthe number
0 of such matrices of size n is
1
n−1
1 (3j+1)!
(1.1) A = =1,2,7,42,429,...
v: n (n+j)!
j=0
i Y
X
After having been a conjecture for severalyears [12], this was first proven by Zeil-
r bergerin [17], and a simpler proofwas given by Kuperberg [9], using a connection
a
with the 6-Vertex Model of statistical mechanics, and an appropriate multivariate
extension of the mere counting function A . A vivid account can be found in [1].
n
It followseasilyfromthe definition thatanalternating signmatrixhas exactly
one +1 on its first (and last) row (and column). Thus we have a sensible four-
variablerefinedstatistics,forthesefourpositionsin 1,...,n 4,togetherwiththeir
{ }
projectionsonasmallernumberofvariables. The dihedralsymmetryofthe square
leaveswithasingleone-variablestatistics(showingaroundformula),andwithtwo
doubly-refined statistics: one, n, for the first and last row (or the rotated case),
Aij
and one, n, for the first row and column (or the three rotated cases), see fig. 1,
Bij
left. Matrices n for n=1,2,3,4,5are given by
A
2000 Mathematics Subject Classification. Primary05E05;Secondary05A15,15A15, 82B23.
Keywordsandphrases. Alternatingsignmatrices,Schurfunctions,Compounddeterminants.
1
2 PHILIPPEBIANE,LUIGICANTINI,ANDANDREASPORTIELLO
i
k
j
Figure 1. Left: a typical alternating sign matrix of size n =
10 (empty cells, disks and diamonds stand respectively for 0, +1
and 1 entries). This matrix contributes to the statistics n and
− Aij
n, with (i,j,k) = (6,4,5). Right: empty cells are replaced by
Bik
scale-shaped tiles, as to produce a valid tiling (i.e., concavities of
neighbouring arcs do match). The direction of the tip specifies if
the cell is of type NW, NE, SE or SW.
0 1 1
0 1
1 = 1 ; 2 = ; 3 = 1 1 1 ;
A A 1 0 A  
(cid:18) (cid:19) 1 1 0
(cid:0) (cid:1)
 
0 7 14 14 7
0 2 3 2
7 21 33 30 14
2 4 5 3  
4 = ; 5 = 14 33 41 33 14 .
A 3 5 4 2 A
14 30 33 21 7 
2 3 2 0  
   7 14 14 7 0 
   
Of course, by definition n = A , i.e. 1,2,7,42,429,... for thecases above.
i,jAij n
Asimplerecursionimpliesthatthesumalongthefirst(andlast)row(andcolumn)
P
gives A , i.e. 1,1,2,7,42,..., and that the bottom-left and top-right entries are
n−1
A , i.e. 1,1,1,2,7,... These simple identities are linear. There exists also qua-
n−2
dratic relations, of Plu¨cker nature, relating these doubly-refined enumerations to
A and the (singly-)refined enumerations (see e.g. [16, 2]).
n
Evaluate now the determinant of these matrices:
det( 2)= 1= 1−1, det( 3)=1=20,
A − − A
det( 4)= 7= 71, det( 5)=1764=422, ...
A − − A
This small numerics suggests a relation that we prove in this paper:
Theorem 1.
(1.2) det( n)=( A )n−3.
n−1
A −
This relation is non-linear. Its degree is not fixed, nor bounded. What is fixed
is what we could call “co-degree”, namely the system size, minus the degree (in
DOUBLY-REFINED ENUMERATION OF ALTERNATING SIGN MATRICES... 3
analogyto the definition of co-dimensionof a subspace). Relations ofthis different
nature seem to be a novelty for the subject at hand.
Our proofofthe theoremabovewill resultas corollaryofa muchmore general
result on certain Schur functions. To see why these two topics are connected, we
have to revert to Kuperberg solution of the Alternating Sign Matrix conjecture.
1.2. ASM, the 6-Vertex Model and Schur functions. It follows from
the connection with the 6-Vertex Model, that the generating function for a certain
weightedenumerationofalternatingsignmatricesisgivenbyacloseddeterminantal
formula. For B = B an ASM, if B = 0, say that (i,j) is a north-west
ij 1≤i,j≤n ij
{ }
(NW) site (resp. NE, SE, SW) if, forgetting the zeroes, the next +1 element along
the same column is in the north direction, and along the same row is in the west
direction (and analogously for the other three cases) – see the right part of fig. 1.
Consider some complex-valued function µ (B) over n n ASMs, and call
n
×
(1.3) Z = µ (B)
n n
B
X
thecorrespondinggeneratingfunction(instatisticalmechanicsµ(B)isageneralized
Gibbs measure – an ordinary measure if it is real-positive and normalized – and Z
is the partition function).
Whenµ (B)hasthefollowingfactorizedform,parametrizedby2n+1variables
n
(x ,...,x ,y ,...,y ,q)=(~x,~y,q),
1 n 1 n
(1.4a) µ (B;~x,~y,q)= w (B);
n i,j
1≤i,j≤n
Y
(q q−1) x y B = 1;
i j ij
− ±
(1.4b) w (B)= q−1x qy B =0, (i,j) is NW or SE;
i,j  i− pj ij
x +y B =0, (i,j) is NE or SW;
 i j ij
−
integrability methods, and a recursion due to Korepin [7], allowed Izergin [6]
to establish a determinantal expression for the generating function Z (~x,~y,q) =
n
µ (B;~x,~y,q). In particular, this function is symmetric under S S acting
B n n× n
on row- and column-parameters x and y .
i j
P
The evaluation of A is recovered if we set q = exp(2πi), x = q−1 for all i
n 3 i
and y =q for all j, as in this case the local weights w become all equal to i√3,
j i,j
regardlessfrom B, and thus µ(B) becomes constant (i.e., the uniform measure, up
to an overall factor).
Lateronithasbeenrecognized[16,14]thatthevalueq =exp(2πi)(sometimes
3
called the combinatorial point) has a special combinatorial property: Z (~x,~y,q)
n
becomes fully symmetric under S (acting on the 2n-uple of qx ’s and q−1y ’s
2n i j
together), more precisely it is proportional to the Schur function associated to
the Young diagram λ = (n 1,n 1,n 2,n 2,...,1,1,0,0), evaluated on
n
− − − −
variables qx ,...,qx ,q−1y ,...,q−1y (see figure 2, left, for a picture of this
1 n 1 n
{ }
Young diagram). One consequence is that we have
(1.5) An =3−(n2)sλn(1,1,...,1),
and also the refined enumerations introduced above are related to specializations
of this Schur function, in which some parameters are left as indeterminates.
4 PHILIPPEBIANE,LUIGICANTINI,ANDANDREASPORTIELLO

2n
Figure2. Left: the Young diagram||{ℓλz′{ℓ}z, }for n = 5. Right: the
n
Young diagram λn,ℓ,ℓ′, for n=5, ℓ=3 and ℓ′ =2.
In particular for the n’s, defining the generating function
Aij
(1.6) (u,v)= nui−1vn−j;
An Aij
1≤i,j≤n
X
one finds
(1.7) An(u,v)=3−(n2)(q2(q+u)(q+v))n−1sλn 1q++quu,1q++qvv,1,...,1 ;
(therationalfunction 1+qu originatesfromthe ratioo(cid:0)fw (B)inthetwol(cid:1)astcases
q+u ij
of (1.4b)).
A detailed analysis of the double-enumeration formula (1.7) restated in terms
ofmultiplecontourintegrals,andtheproofofarelationwithadouble-enumeration
formula for totally-symmetric self-complementary plane partitions in a hexagonal
box of size 2n, can be found in [5].
1.3. On the determinants of Schur functions. In this section we state a
theoremconcerningthedeterminantofamatrixwhoseelementsareSchurfunctions
s . Not surprisingly, as these functions are related to ASM enumerations e.g.
λn
through equations (1.5) and (1.7), this property will show up to be the structure
behind Theorem 1, and conceivably, it has an interest by itself. For this reason, in
this paper we pursue the task of stating and proving a much wider version of the
forementionedproperty,than the one that would suffice for Theorem1. This leads
us to introduce a wider family of Young diagrams.
We define the 2-staircase diagram λn,ℓ,ℓ′, for n 1, 0 ℓ′ ℓ, as
≥ ≤ ≤
(1.8) λn,ℓ,ℓ′ = (n 1)ℓ+ℓ′,(n 1)ℓ,(n 2)ℓ+ℓ′,(n 2)ℓ,...,ℓ′,0
− − − −
i.e. (λn,ℓ,ℓ′)2j−1 = (n(cid:0) j)ℓ+ℓ′ and (λn,ℓ,ℓ′)2j = (n j)ℓ (see figure 2, rig(cid:1)ht). We
− −
call the associated Schur polynomial, s , a 2-staircase Schur function.
λn,ℓ,ℓ′
The name comes from the fact that this family of diagrams generalizes the
well-known family of staircase diagrams µ
n,ℓ
(1.9) µ = (n 1)ℓ,(n 2)ℓ,...,ℓ,0 .
n,ℓ
− −
The Schur functions sλn are t(cid:0)hus particular cases of 2-s(cid:1)taircase Schur functions,
corresponding to ℓ=1 and ℓ′ =0.
The polynomials s have been considered recently by Alain Lascoux. In
λn,ℓ,ℓ′
particular,in[11, Lemma13]they areshownto coincide with the specializationat
q =exp(2πi) of a certain natural extension of Gaudin functions.
ℓ+2
DOUBLY-REFINED ENUMERATION OF ALTERNATING SIGN MATRICES... 5
In an apparently unrelated context we see the appearence of the polynomials
s , for ℓ′ = 0 only. This context, analysed by Paul Zinn-Justin in [18], is the
λn,ℓ,ℓ′
study of the solution of the qKZ equation related to the spin ℓ/2 representation of
thequantumaffinealgebraU (sl(2))withq =exp(2πi). Itisshownthat,bytaking
q ℓ+2
thescalarproductofthesolutionoftheqKZequationwithanaturalreferencestate,
one obtains s . d
λn,ℓ,0
As anticipated, our Theorem 1 will be a corollary of the following result, of
independentinterest,whichexhibits aremarkablefactorizationofadeterminantof
2-staircase Schur functions:
Theorem 2. Let N = ℓ(n 1)+ ℓ′ + 1. Let x ,y be indeterminates,
i i 1≤i≤N
− { }
let f(~z,w ,w ) stand for f(z ,...,z ,w ,w ), and, for an ordered N-uple ~x =
1 2 1 2n−2 1 2
(x ,x ,...,x ), let ∆(~x)= (x x ) denote the usual Vandermonde determi-
1 2 N i<j i− j
nant. Then
Q
det s (~z,x ,y )
λn,ℓ,ℓ′ i j 1≤i,j≤N
(1.10) (cid:16) (cid:17) 2n−2
=c(n,ℓ,ℓ′)∆(~x)∆(~y) zℓ′(ℓ+1) sℓ (~z)sℓ(n−2)+ℓ′−1(~z).
i µ2n−2,ℓ+1 λn−1,ℓ,ℓ′
(cid:18) i=1 (cid:19)
Y
The quantity c(n,ℓ,ℓ′) is valued in 0, 1 . More precisely,
{ ± }
(1.11) c(n,ℓ,ℓ′)= (−1)(n−1)(ℓ+21)+(ℓ′2+1) n=1 or gcd(ℓ+2,ℓ′+1)=1
( 0 n>1 and gcd(ℓ+2,ℓ′+1)=1
6
Remark that, as well known, the staircase Schur function s can be further
µ2n−2,ℓ+1
factorized. Let us recall the definition of the (bivariate homogeneous) Chebyshev
polynomials (of the second kind)
xh+1 yh+1
(1.12) U (x,y)= − =xh+xh−1y+ +yh.
h
x y ···
−
One can write (cf. equation (A.7))
(1.13) s (~z)= U (z ,z ).
µN,h h i j
1≤i<j≤N
Y
As Schur functions have several determinant representations (see Appendix A),
the left-hand–side quantity of the theorem is a “determinant of determinants”, a
structure in linear algebra that is sometimes called a compound determinant [13,
ch. VI]. As we will see, the theory of compound determinants will have a crucial
role in our proof.
ResultsintheformofTheorem2,oratleastapproachestoquantitiesasinthe
left-hand side of equation (1.10), already exist in the literature, although mostly
with partitions of comparatively simpler structure. Cf. [11], where also a gen-
eral approach is outlined. In particular, equations (23) and (24) of [11] have a
form of striking similarity with our theorem above, while involving respectively a
rectangularpartitionrp (r,r,...,r)(p times), andthe basic1-staircasepartition
≡
(r,r 1,r 2,...,1,0),andtheunnumberedthirdequationafterCorollary9of[11]
− −
(for which, however,no factorizationis stated) has a similar structure to what will
bethe matrixofouranalysis,withthe onlydifferencethatitpresentsaChebyshev
polynomial at the denominator instead that the numerator.
6 PHILIPPEBIANE,LUIGICANTINI,ANDANDREASPORTIELLO
Theorem2iseasilyseentoholdatn=1andany(ℓ,ℓ′). Thiscouldseemagood
baseforaninduction. Howeverweuseinductiveargumentsonlyfortheminortask
of determining the overall constant c(n,ℓ,ℓ′), in section 4.2. Conversely,in section
4.1 we prove divisibility results, by a method reminiscent of the “exhaustion of
factors” described in Krattenthaler’s survey [8].
Notehoweverthatthefactorss arepolynomialsof‘large’degree,ℓn(n
λn−1,ℓ,ℓ′ −
1)+ℓ′n,withnofactorizationsaslongasgcd(ℓ+2,ℓ′+1)=1(wegiveapartialproof
ofthisstatementinProposition5below–afullproofisnothardtoachieve). Thus,
in a sense, the tools we develop in section 3 should be regarded as an extension of
the exhaustion of factor method to the case in which we have an infinite family of
determinantalidentities,andsomeofthefactorshaveanunbounded degree,scaling
with the size parameter associated to the family.
Finally,letusaddafew wordsonnotations: alongthe paper,if~z is avectorof
length n (the length will be clear by the context), we write f(~z) as a shortcut for
f(z ,...,z ), and f(~z,w ,w ,...) as a shortcut for f(z ,...,z ,w ,w ,...). We
1 n 1 2 1 n 1 2
also write and f(~zri1···ik,w1,w2,...) if the variables zi1, ..., zik are dropped from
the list (z ,...,z ).
1 n
Thepaperisorganizedasfollows. Insection2weshowhowtoderiveTheorem1
from Theorem 2 specialized to ℓ = 1 and ℓ′ = 0. In section 3 we present some
preparatory lemmas to the proof of Theorem 2, which is presented in section 4.
Appendix A collects some basic definitions and facts on Schur functions, while in
appendixBweintroduceanevenlargerclassofstaircaseSchurfunctions,andstudy
some of their properties.
2. Derivation of Theorem 1 from Theorem 2
For a polynomial f(x,y), denote by f(x,y) the coefficient of the monomial
|[xiyj]
xiyj. We first state a simple but useful lemma.
Lemma 1. Let P(u,v) be a polynomial in two indeterminates, of degree at most
n−1 in each variable. Call P = (P(u,v)|[ui−1vj−1])1≤i,j≤n. Let ui, vj be indeter-
minates, then
(2.1) det P(u ,v ) =∆(~u)∆(~v) detP .
i j 1≤i,j≤n
Proof. Call V(~u) (cid:0)the Vande(cid:1)rmonde matrix V = uj−1. Then detV(~u) =
ij i
∆(~u), and the matrix P(u ,v ) is the product V(~u)TP V(~v). (cid:3)
i j 1≤i,j≤n
(cid:0) (cid:1)
This lemma allows us to state that our Theorem 2 is equivalent to
det s (~z,x,y) =
λn,ℓ,ℓ′ |[xiyj] 0≤i,j≤ℓ(n−1)+ℓ′
(2.2) (cid:16) 2n−2 (cid:17)
=c(n,ℓ,ℓ′) zℓ′(ℓ+1) sℓ (~z)sℓ(n−2)+ℓ′−1(~z),
i µ2n−2,ℓ+1 λn−1,ℓ,ℓ′
(cid:18) i=1 (cid:19)
Y
(of course, with c(n,ℓ,ℓ′) as in (1.11)).
Now we proceed to the proof of Theorem 1. One can compute, with ~u =
(u ,...,u ),
1 n
(2.3) ∆ 1q++quuii =∆(~u)(q2−1)(n2) (q+ui)−(n−1).
i
(cid:0)(cid:8) (cid:9)(cid:1) Y
DOUBLY-REFINED ENUMERATION OF ALTERNATING SIGN MATRICES... 7
It follows from Lemma 1, and equation (1.7), that
(2.4)
∆(~u)∆(~v)det( n)=
Aij
=(−1)(n2)det(cid:18)(q2(q+ui3)((nq2)+vj))n−1sλn 1q++quuii,1q++qvvjj,1,...,1 (cid:19)1≤i,j≤n
(cid:0) (cid:1)
= −q4 (n2) n (q+u )(q+v ) n−1det s 1+qui,1+qvj,1,...,1 .
3n i i λn q+ui q+vj 1≤i,j≤n
(cid:18) (cid:19) iY=1(cid:0) (cid:1) (cid:16) (cid:0) (cid:1)(cid:17)
Using Theorem 2 with ℓ=1 and ℓ′ =0 on the determinant on the right-hand side
(with x = 1+qui and y = 1+qvj), and then (2.3), we obtain
i q+ui j q+vj
(n)
∆(~u)∆(~v)det( n)=∆(~u)∆(~v)( 1)n−1+(n2) (q−q2)2 2
(2.5) Aij − 3n
(cid:18) (cid:19)
s (1,1,...,1)sn−3 (1,1,...,1)
× µ2n−2,2 λn−1
Recognize that (q q2)2 = 3. By the explicit evaluation of a staircase Schur
− −
function, equation (1.13), we have
(2n−2)
(2.6) sµ2n−2,1(1,1,...,1)=3 2
Theorem 1 follows from (1.5), (2.5), (2.6). (cid:3)
3. Preliminary results
3.1. On the minor expansion of a sum of matrices. Consider k n n
×
(a)
matrices of indeterminates M , 1 i,j n; 1 a k. For I,J [n], denote by
ij ≤ ≤ ≤ ≤ ⊆
M therestrictionofM torowsinI andcolumnsinJ. Denoteby =(I ,...,I )
I,J 1 k
I
an ordered k-uple of subsets I [n] (possibly empty), forming a partition of [n].
a
⊆
For two such k-uples and , say that they are compatible if I = J for all
a a
I J | | | |
a=1,...,k, and write in this case. Denote by ǫ( , ) the signature of the
I ∼J I J
permutation that reorders (I ,...,I ) into (J ,...,J ), with elements within the
1 k 1 k
blocks in order. Then we have
Proposition 1 (Minor expansion of a sum of matrices).
k k
(3.1) det M(a) = ǫ( , ) detM(a) .
I J Ia,Ja
(cid:16)Xa=1 (cid:17) XI,J aY=1
I∼J
Proof. Consider the full expansion of the determinant
k n k
det M(a) = ǫ(σ) M(a)
iσ(i)
(3.2) (cid:16)Xa=1 (cid:17) σX∈Sn iY=1(cid:16)Xa=1 (cid:17)
n
= ǫ(σ) M(b(i))
iσ(i)
σX∈Snb∈X[k]n iY=1
Associate to each pair (σ,b) in the linear combination above, a pair ( , ) of com-
I J
patible partitions, through
(3.3) I = i : b(i)=a ; J = j : b(σ−1(j))=a .
a a
{ } { }
8 PHILIPPEBIANE,LUIGICANTINI,ANDANDREASPORTIELLO
So is determined by b alone, and all the permutations σ producing the same
I J
can be written as the “canonical” permutation τ, that reorders (I ,...,I ) into
1 k
(J ,...,J ) with elements within the blocks in order, acting from the left on a
1 k
permutation ρ = ρ S S . The signature factorizes, ǫ(σ) =
a a ∈ I1 × ··· × Ik
ǫ(τ) ǫ(ρ ), and ǫ(τ)=ǫ( , ) by definition, thus
a a Q I J
Q k
det M(a) = ǫ( , ) ǫ(ρ ) M(a)
(3.4) I J a iτ◦ρa(i)
(cid:16)Xa=1 (cid:17) XI,J Ya ρaX∈SIa iY∈Ia
I∼J
For each index a, the sum over the permutations ρ produces the appropriate
a
determinant of the minor. (cid:3)
3.2. Bazin-Reiss-Picquet Theorem. In this section we recall the Bazin-
Reiss-Picquet Theorem [13, pg. 193-195, 202-204].
§
Take a triplet of integers m n p 0. Call S the set of subsets of [n], of
n,p
≥ ≥ ≥
cardinality p (thus S = n ). For a set I S , write I = i ,...,i for the
| n,p| p ∈ n,p { 1 p}
ordered list of elements.
(cid:0) (cid:1)
Consider the m n matrices of indeterminates A and B, and the m (m n)
× × −
matrix of indeterminates C. Write (X Y) for the matrix resulting from taking all
|
the columns of X, followed by all the columns of Y.
For a pair (I,J) S S define MI,J as the matrix
n,p n,p
∈ ×
A k n, k I;
h,k
(3.5) MI,J = B k ≤=i ; 6∈
h,k  h,jℓ ℓ
C n<k m;
 h,k−n
≤
(thatis,replacethe columnsI of(AC) withthe columnsJ ofB,inorder). Define
|
D = detMI,J. Choose a total ordering of S , and construct the matrix D =
I,J n,p
D , of dimension n . Then the compound determinant detD does not
I,J I,J∈Sn,p p
depend on the chosen ordering, and has the following factorization property:
(cid:0) (cid:1) (cid:0) (cid:1)
Theorem 3 (Bazin-Reiss-Picquet).
(n−1) (n−1)
(3.6) detD =det(AC) p det(B C) p−1 .
| |
3.3. A divisibility corollary. A corollary of the Bazin-Reiss-Picquet Theo-
remisadivisibilityresultforaspecialfamilyofdeterminants. Takem n k 0.
≥ ≥ ≥
Considermindeterminatesz ,nindeterminatesy ,and2nk indeterminatesua, va,
i j i i
with 1 i n and 1 a k (ua, va may possibly be elements in the polynomial
≤ ≤ ≤ ≤ i i
ring R(z,y)). Take m polynomial functions f (x), and introduce the associated
j
Slater determinant, that is, the totally-antisymmetric polynomial
(3.7) P(~x)=P(x ,...,x )=det f (x ) .
1 m j i 1≤i,j≤m
AtypicalexamplecouldbeashiftedVandermon(cid:0)de,P(x(cid:1),...,x )=∆ (x ,...,x )
1 m λ 1 m
for λ a partition of length m (see appendix A).
Then we have
Proposition2. Thepolynomial det ka=1uaivja P(~zri,yj) is divisible by
1≤i,j≤n
n−k (cid:16) (cid:17)
the polynomial P(~z) . P
(cid:0) (cid:1)
DOUBLY-REFINED ENUMERATION OF ALTERNATING SIGN MATRICES... 9
Proof. Apply the formula for the minor expansion of a sum of matrices,
Proposition 1, to get
k
det uaivjaP(~zri,yj)
1≤i,j≤n
(cid:16)Xa=1 (cid:17)
(3.8)
k
= ǫ(I,J) uai vja det P(~zri,yj) i∈Ia,j∈Ja.
I,J 1≤a≤k 1≤a≤k a=1
IX∼J iY∈Ia jY∈Ja Y (cid:0) (cid:1)
Now apply the Bazin-Reiss-Picquet Theorem to each of the determinants, with
(m,n,p) (m, I ,1), and get
α
→ | |
(3.9) det P(~zri,yj) i∈Ia,j∈Ja =P(~z)|Ia|−1P(~zrIa,~yr(Ja)c).
Thus we have (cid:0) (cid:1)
k
det uaivjaP(~zri,yj)
1≤i,j≤n
(cid:16)Xa=1 (cid:17)
(3.10)
k
=P(~z)n−k ǫ(I,J) uai vja P(~zrIa,~yr(Ja)c);
I,J 1≤a≤k 1≤a≤k a=1
X Y Y Y
I∼J i∈Ia j∈Ja
and the quantity in the sum on the right-hand side is a polynomial. (cid:3)
3.4. Vanishing and recursion properties of 2-staircase Schur func-
tions. Here we gather some relevant facts about the family of 2-staircase Schur
functions s (~z) introduced in (1.8). In this section we use q as a synonym of
λn,ℓ,ℓ′
exp(2πi).
ℓ+2
Proposition 3 (wheel condition). For distinct g, h and k in 0,...,ℓ+1 , and
{ }
distinct i, j and m in 1,...,2n ,
{ }
(3.11) sλn,ℓ,ℓ′(~zrijm,qgw,qhw,qkw)=0.
Proposition4 (recursionrelation). Fork in 1,...,ℓ+1 ,andi,j in 1,...,2n ,
{ } { }
distinct,
(3.12) sλn,ℓ,ℓ′(~zrij,w,qkw)=wℓ′Uℓ′(1,qk) Uzℓ+1(zmqk,ww) sλn−1,ℓ,ℓ′(~zrij).
m
1≤m≤2n −
Y
m6=i,j
Propositions 3 and 4 are occurrences, already known in the literature (cf. e.g. [18,
Thm.4]),ofvanishing conditions(andrelatedrecursionproperties)within a broad
family,forwhichthename“wheelcondition”isoftenused. Therehasbeenarecent
interestinthe investigationofthe structureofthe correspondingideals,inthe ring
of symmetric polynomials (see e.g. [3, 4]).
We provethe propositionsaboveinAppendix B. Moreprecisely,inthe appen-
dixwegeneralize2-staircaseSchurfunctionstothem-staircase case,andprovethe
appropriate generalizations of the propositions above, together with some further
properties of potential future interest.
Notice that, if gcd(ℓ′+1,ℓ+2)=g >1, then there exists some 1 k ℓ+1
≤ ≤
such that qk is a root of Uℓ′(1,x) (e.g., k = (ℓ + 2)/g). Then it follows from
10 PHILIPPEBIANE,LUIGICANTINI,ANDANDREASPORTIELLO
equation(3.12)thats vanishesifz =qkz ,i.e.itisdivisible byz qkz . On
λn,ℓ,ℓ′ i j i− j
the contrary,if gcd(ℓ′+1,ℓ+2)=1, one has the following proposition
Proposition 5. Suppose gcd(ℓ′ +1,ℓ+2) = 1 and n 2, then s has no
≥ λn,ℓ,ℓ′
factors of the form (z ηz ), for any 1 i,j 2n and η C.
i j
− ≤ ≤ ∈
Proof. We prove the statement by induction on n. The case n = 2 is done
by direct inspection of s 1. Now suppose the statement true up to n 1
λn,ℓ,ℓ′ −
and assume that there exists i,j 1,...,2n and η C such that (z ηz )
i j
divides s . Then take k and∈h {distinct in}dices in∈1,...,2n r i,j −(note
λn,ℓ,ℓ′ { } { }
that we need n 2 at this point), and specialize s . The linear term
≥ λn,ℓ,ℓ′|zk=qzh
z ηz must divide also the specialized polynomial, and, using the recursionrela-
i j
−
tion of Proposition 4, it must divide the corresponding right-hand–side expression
for (3.12). However, this expression is non-zero for the other variables z being
m
generic (because the only potentially dangerous factor, Uℓ′(1,qk), may vanish only
if gcd(ℓ′ +1,ℓ+2) > 1), and the factors of the form zℓ′, and U (z ,z ), for
k ℓ+1 m k
m=k,h, do not contain z ηz as a factor. Thus z ηz must divide s ,
thi6s being in contrast withit−he injductive assumption.i− j λn−1,ℓ,ℓ(cid:3)′
4. Proof of Theorem 2
As outlined in the introduction, our strategy for proving Theorem 2 will be as fol-
lows: let us call ψn,ℓ,ℓ′(z,x,y) the left-hand side of (1.10); first we identify several
polynomial factors of ψn,ℓ,ℓ′(z,x,y); then we show that these factors are relatively
primeandthattheirproductexhauststhedegreeofψn,ℓ,ℓ′(z,x,y);finally,wedeter-
mine the overallconstantfactor. As in the previous subsection, also in this section
2πi
we set q =eℓ+2.
4.1. Polynomial factors of ψn,ℓ,ℓ′(~z,~x,~y). Westartbyidentifyingapolyno-
mialfactor ofψn,ℓ,ℓ′(~z,~x,~y)whosefactorizationinvolvesonlymonomialsandbino-
mials. By virtue of Lemma 1, we have that ψn,ℓ,ℓ′(~z,~x,~y) is divisible by ∆(~x) and
∆(~y). Since the degreeofψn,ℓ,ℓ′ ineachvariablexi or yi separatelyis(n 1)ℓ+ℓ′,
−
whichisthesameasthedegreeof∆(~x)∆(~y),thequotientisapolynomialofdegree
zero in x and y (namely, it is the determinant of the matrix of coefficients in x
i j
and y of sλn,ℓ,ℓ′(~z,x,y)). Call Qn,ℓ,ℓ′(~z) the resulting quotient
ψn,ℓ,ℓ′(~z,~x,~y)
(4.1) Qn,ℓ,ℓ′(~z)=
∆(~x)∆(~y)
We work out immediately the case of Theorem 2 corresponding to the second case
of equation (1.11)
Proposition 6. If gcd(ℓ′+1,ℓ+2)>1 and n 2, then Qn,ℓ,ℓ′(~z)=0
≥
1E.g., realize that, for z1−ηz2 to divide the Schur function, it should divide the shifted
Vandermondeatnumerator,withahigherpowerw.r.t.theordinaryVandermondeatdenominator.
The case η = 1 is easily ruled out (even if we further specialize z3 = z, z4 = 0, we obtain
sλ2,ℓ,ℓ′(z,z,z,0)=z2(ℓ+ℓ′)(ℓ+2)(ℓ′+1)(ℓ−ℓ′+1)/2, which is not identically zero as we have
ℓ ≥ 0 and 0 ≤ ℓ′ ≤ ℓ). For η 6= 1 we can have no simplifications with the Vandermonde at
denominator,anditsufficestoanalysetheshiftedVandermonde,whichgives
∆λ2,ℓ,ℓ′(z,ηz,0,1)=zℓ+ℓ′+3(cid:0)((ηz)ℓ+2−1)(ηℓ′+1−1)−((ηz)ℓ′+1−1)(ηℓ+2−1)(cid:1).
Again,thisisnotidenticallyzero,as,forthegcdhypothesis,ηℓ′+1−1andηℓ+2−1cannotvanish
simultaneously.