On the Existence of Tableaux with Given Modular Major Index

0.24 MB·

by  Joshua P. Swanson

ON THE EXISTENCE OF TABLEAUX WITH GIVEN
MODULAR MAJOR INDEX
7
1
JOSHUA P. SWANSON
0
2
n Abstract. Weprovidesimplenecessaryandsufficientconditions
a fortheexistenceofastandardYoungtableauofagivenshapeand
J major index r mod n, for all r. Our result generalizes the r = 1

8 casedueessentiallytoKlyachko[Kly74]andprovesarecentconjec-
1
ture due to Sundaram [Sun16] for the r =0 case. A byproduct of

the proof is an asymptotic equidistribution result for “almost all”
]
O shapes. The proof uses a representation-theoretic formula involv-
C ing Ramanujan sums and normalized symmetric group character
. estimates. Furtherestimatesinvolving“opposite”hooklengthsare
h
given which are well-adapted to classifying which partitions λ n
at have fλ nd for fixed d. We also give a new proof of a gene⊢ral-
m ≤
ization of the hook length formula due to Fomin-Lulov [FL95] for
[ symmetric groupcharactersatrectangles. We conclude withsome

remarks on unimodality of symmetric group characters.
1
v
3
6
9
4 1. Introduction
0
.
1 WeassumebasicfamiliaritywiththecombinatoricsofYoungtableaux
0 and the representation theory of the symmetric group. For further in-
7
formation and definitions, see [Ful97], [Sta99], or [Sag01].
1
: Let λ n be an integer partition of size n, and let SYT(λ) denote
v
⊢
i the set of standard Young tableaux of shape λ. We write λ′ for the
X
transpose (or conjugate) of λ. Let majT denote the major index of
r
a T SYT(λ). We are chiefly interested in the constants
∈
a := # T SYT(λ) : majT r
λ,r n
{ ∈ ≡ }
where r is taken mod n. To avoid giving undue weight to trivial
cases, we take n 1 throughout. Work due to Klyachko and, later,
≥
Kraskiewicz-Weyman, gives the following:
Date: January 19, 2017.
Key words and phrases. standard Young tableaux, symmetric group characters,
major index, hook length formula, rectangular partitions.
The author was partially supported by the National Science Foundation grant
DMS-1101017.
1
2 JOSHUA P.SWANSON
Theorem 1.1 ([Kly74, Proposition 2], [KW01]). Let λ n and n 1.
⊢ ≥
The constant a is positive except in the following cases, when it is
λ,1
zero:
λ = (2,2) or λ = (2,2,2);
•
λ = (n) when n > 1; or λ = (1n) when n > 2.
•
Indeed, the constants a can be interpreted as representation multi-
λ,r
plicities as follows, a result originally due to Kraskiewicz-Weyman. Let
C be the cyclic group of order n generated by the long cycle σ :=
n n
(12 n) S , let Sλ be the Specht module of shape λ n, and let
n
χr:·C·· ∈C× be the irreducible representation given by χ⊢r(σi) := ωri
wherenω→is a fixed primitive nth root of unity and r Z/n. Lent , n
n
∈ h− −i
denote the standard scalar product for complex representations.
Theorem 1.2 (see [KW01, Theorem 1]). With the above notation, we
have
Sλ,χr Sn = a = χr,Sλ Sn .
h ↑Cni λ,r h ↓Cni
Moreover, a depends only on λ and gcd(n,r).
λ,r
Remark 1.3. Kraskiewicz-Weyman gave the first equality in 1.2, and
the second follows by Frobenius reciprocity. Klyachko [Kly74, Proposi-
tion2] actually determined which Sλ contain faithful representations of
C in agreement with 1.1. One may see through a variety of methods
n
that χr Sn depends up to isomorphism only on gcd(r,n).
↑Cn
Themanuscript[KW01]waslong-unpublished, thedelaybeinglargely
due to Klyachko having already given a significantly more direct proof
of their main application, relating χ1 Sn to free Lie algebras, though
↑Cn
we have no need of this connection. For a more modern and readable
account of these results, see [Reu93, Theorems 8.8-8.12].
The following conjecture due to Sundaram was originally stated in
terms of the multiplicity of Sλ in 1 Sn.
↑Cn
Conjecture 1.4. [Sun16]. Let λ n and n 1. The constant a is
λ,0
⊢ ≥
positive except in the following cases, when it is zero: n > 1 and
λ = (n 1,1)
• −
λ = (2,1n−2) when n is odd
•
λ = (1n) when n is even.
•
1.4 is the r = 0 case of our main result:
Theorem 1.5. Let λ n and n 1. The constant a is positive
λ,r
⊢ ≥
except in the following cases, when it is zero: n > 1 and
λ = (2,2), r = 1,3; or λ = (2,2,2), r = 1,5; or λ = (3,3),
•
r = 2,4;
TABLEAUX WITH MODULAR MAJOR INDEX 3
λ = (n 1,1) and r = 0;
• −
0 if n is odd
λ = (2,1n−2), r =
• n if n is even;
(
2
λ = (n), r 1,...,n 1 ;
• ∈ { − }
1,...,n 1 if n is odd
λ = (1n), r { − }
• ∈ 0,...,n 1 n if n is even.
(
{ − }−{2}
Equivalently, using 1.2, every irreducible representation appears in
each χr Sn or Sλ Sn except in the noted exceptional cases.
↑Cn ↓Cn
Our main tool is the following representation-theoretic formula due
toD´esarm´enien. Let χλ(µ)denotethecharacter ofSλ atapermutation
of cycle type µ. Write fλ := χλ(1n) = dimSλ = #SYT(λ).
Theorem 1.6. [D´es90, Th´eor`eme 2.2] Let λ n and n 1. For all
r Z/n, ⊢ ≥
∈
a 1 1 χλ(ℓn/ℓ)
λ,r
= + c (r)
fλ n n fλ ℓ
ℓ|n
X
ℓ6=1
where
ℓ φ(ℓ)
c (r) := µ
ℓ
gcd(ℓ,r) φ(ℓ/gcd(ℓ,r))
(cid:18) (cid:19)
is a Ramanujan sum, µ is the classical M¨obius function, and φ is Eu-
ler’s totient function.
We estimate the quotients in the preceding formula using the follow-
ing result due to Fomin and Lulov. A ribbon is a connected skew shape
with no 2 2 rectangles.
×
Theorem 1.7. [FL95, Theorem 1.1] Let λ n where n = ℓs. Then
⊢
s!ℓs
χλ(ℓs) (fλ)1/ℓ.
| | ≤ (n!)1/ℓ
Indeed, 1.7 is based on the following generalization of the hook
length formula (the ℓ = 1 case), which seems less well-known than it
deserves. We give an alternate proof of 1.8 in Section 5 along with
further discussion. For λ n, write c λ to mean that c is a cell in λ.
⊢ ∈
Further write h for the hook length of c and write [n] := 1,2,...,n .
c
{ }
4 JOSHUA P.SWANSON
Theorem 1.8 ([FL95, Corollary 2.2]; see also [JK81, 2.7.32]). Let λ
⊢
n where n = ℓs. Then
i
i∈[n]
(1) χλ(ℓs) = i≡Qℓ0
| | h
c
c∈λ
hcQ≡ℓ0
whenever λ can be written as s successive ribbons of length ℓ (i.e. when-
ever the “ℓ-core” of λ is ∅), and 0 otherwise.
Other work on q-analogues of the hook length formula has focused
on algebraic generalizations and variations on the hook walk algorithm
rather than evaluations of symmetric group characters. For instance,
anapplicationofKerov’sq-analogueofthehookwalkalgorithm[Ker93]
was to prove a recursive characterization of the right-hand side of (4)
below. See [CFKP11, 6] for a relatively recent overview of literature
§
in this direction.
We also give the following asymptotic uniform distribution result
which largely strengthens 1.5.
Theorem 1.9. Let λ n be a partition where fλ n5 1. Then for
⊢ ≥ ≥
all r,
a 1 1
λ,r
< .
fλ − n n2
(cid:12) (cid:12)
In particular, if n 81, λ (cid:12)< n 7,(cid:12)and λ′ < n 7, then fλ n5 and
≥ 1(cid:12) − (cid:12) 1 − ≥
the inequality holds. (cid:12) (cid:12)
Indeed, the upper bound in 1.9 is quite weak and is intended only to
convey the flavor of the distribution of (a )n−1 for fixed λ. One may
λ,r r=0
use Roichman’s asymptotic estimate [Roi96] of χλ(ℓs) /fλ to prove
| |
exponential decay in many cases. Moreover, one typically expects fλ
to grow super-exponentially, i.e. like (n!)ǫ for some ǫ > 0 (see [LS08]
for some discussion and a more recent generalization of Roichman’s
result), which in turn would give a super-exponential decay rate in 1.9.
We have no need for such refined statements and so have not pursued
them further.
The rest of the paper is organized as follows. In Section 2, we recall
earlier work. In Section 3 we discuss and generalize 1.6. In Section 4,
we use symmetric group character estimates and a new estimate involv-
ing “opposite hook products,” 4.4, to deduce our main results, 1.5
and 1.9. We give an alternative proof of 1.8 in Section 5. In Section 6,
we briefly discuss unimodality of symmetric group characters in light
of 4.4.
TABLEAUX WITH MODULAR MAJOR INDEX 5
2. Background
Here we review objects famously studied by Springer [Spr74, (4.5)]
and Stembridge [Ste89] and give further background for use in later
sections. All representations will be finite-dimensional over C.
Continuing ourearlier notation, λ nisapartitionofsize n, SYT(λ)
⊢
is the set of standard Young tableaux of shape λ and which has cardi-
nality fλ, (12 n) is the long cycle in the symmetric group S , Sλ is
n
···
the irreducible S -module (Specht module) of shape λ with character
n
at an element of cycle type µ given by χλ(µ), c λ denotes a cell in
∈
the Ferrers diagram of λ, and h denotes the hook length of that cell.
c
Let G be a finite group, g G a fixed element of order n, M a
∈
finite dimensional G-module, and ω a fixed primitive nth root of unity.
n
Suppose ωe1,ωe2,... is the multiset of eigenvalues of g acting on
{ n n }
M. The multiset e ,e ,... lists the cyclic exponents of g on M;
1 2
{ }
these integers are well-defined mod n. Following [Ste89], define the
corresponding “modular” generating function as
P (q) := qe1 +qe2 + (mod (qn 1)).
M,g
··· −
Write χM(g) to denote the character of M at g. Note that
(2) P (ωs) = χM(gs),
M,g n
so that for instance P (q) depends only on the conjugacy class of g.
M,g
When G = S and g S has cycle type µ n, we write P (q) :=
n n M,µ
∈ ⊢
P (q).
M,g
Theorem 2.1 (see [Ste89, Theorem 3.3] and [KW01]). Let λ n. The
⊢
cyclic exponents of (12 n) on Sλ are the major indices of SYT(λ),
···
mod n, and
(3) P (q) qmajT
Sλ,(n)
≡
T∈SYT(λ)
X
a  qi (mod (qn 1)).
λ,r
≡ −
r|n 1≤i≤n
X  X 
gcd(i,n)=r
 
 
Remark 2.2. Stembridgegavethefirstequalityin 2.1. Equalityofthe
first and third terms follows immediately from Kraskiewicz-Weyman’s
work using 1.2 and the observation that the multiplicity of χr in Sλ Sn
↓Cn
is the number of times r appears as a cyclic exponent of (12 n) in
···
Sλ.
We also recall Stanley’s q-analogue of the hook length formula:
6 JOSHUA P.SWANSON
Theorem 2.3. [Sta99, 7.21.5] Let λ n with λ = (λ ,λ ,...). Then
1 2
⊢
qd(λ)[n] !
(4) qmaj(T) = q
[h ]
T∈SYT(λ) c∈λ c q
X
where d(λ) := (i 1)λ , [n] ! := [nQ] [n 1] [1] , and [a] :=
i q q q q q
1+q + +qa−1 = −qa−1. − ···
··· P q−1
The representation-theoretic interpretation of the coefficients a in
λ,r
1.2 is related to the following result due independently to Lusztig (un-
published) andStanley. Werecordittogiveourresultscontext, though
it will not be used in our present work. For λ n and i Z, define
⊢ ∈
b := # T SYT(λ) : majT = i
λ,i
{ ∈ }
so that b = a .
k∈Z λ,i+kn λ,i
TheorePm 2.4. [Sta79, Proposition 4.11] Let λ n. The multiplicity
⊢
of Sλ in the ith graded piece of the type A coinvariant algebra is b .
n−1 λ,i
Indeed, thesecondequalityin 1.2followsfrom 2.4and[Spr74,Prop.4.5].
See also [ABR05, p. 3059] for a more recent refinement of 2.4 andsome
further discussion.
Finally, we have need of the so-called Ramanujan sums.
Definition 2.5. Given j Z and s Z, the corresponding Ramanu-
>0
∈ ∈
jan sum is
c (s) := the sum of the sth powers of the primitive jth roots of unity.
j
For instance, c (2) = i2 + ( i)2 = 2 = µ(4/2)φ(4)/φ(2). The
4
− −
equivalence of this definition of c (s) and the formula in 1.6 is classical
j
and was first given by H¨older; see [Kno75, Lemma 7.2.5] for a more
modern account. These sums satisfy the well-known relation
n r = s
(5) c (n/s)c (n/v) =
v r
0 r = s
(
v|n 6
X
for all s,r n [Kno75, Lemma 7.2.2].
|
3. Generalizing the Main Formula
In this section we discuss 1.6 and present a straightforward gener-
alization. We begin with a proof of 1.6 similar to but different from
that in [D´es90]. It is included chiefly because of its simplicity given the
background in Section 2 and because part of the argument will be used
below in Section 5.
TABLEAUX WITH MODULAR MAJOR INDEX 7
Proof of 1.6. Pick s n, so (12 n)s has cycle type ((n/s)s). Evalu-
| ···
ating (3) at q = ωs gives
n
(6) χλ((n/s)s) = P (ωs) = a c (s)
Sλ,(n) n λ,r n/r
r|n
X
since (ωs)i = (ωi)s and ωi is a primitive n/gcd(i,n)th root of unity.
n n n
We may now consider (6) as a system of linear equations indexed by
s n in variables a indexed by r n with coefficient matrix C :=
λ,r
| |
(c (s)) . For example, the s = n linear equation reads
n/r s|n,r|n
fλ = χλ(1n) = a φ(n/r),
λ,r
r|n
X
which follows immediately from the fact that fλ = n−1a and that
r=0 λ,r
a depends only on gcd(r,n).
λ,r
P
As it happens, the coefficient matrix C is nearly its own inverse.
Precisely,
(7) (c (s))2 = nI,
n/r s|n,r|n
where I is the identity matrix with as many rows as positive divisors
of n. It is easy to see that (7) is equivalent to the identity (5) above.
Using (7) to invert (6) gives
a n = χλ((n/s)s)c (r).
λ,r n/s
s|n
X
For the s = n term, we have c (r) = 1 and χλ(1n) = fλ. Tracking this
1
term separately, dividing by n and replacing s with ℓ := n/s now gives
1.6, completing the proof. (cid:3)
Variations on 1.6 have appeared in the literature numerous times
in several guises. The corresponding symmetric function expansion is
older and due to H. O. Foulkes. Let ch denote the Frobenius charac-
teristic map and let p denote the power symmetric function indexed
λ
by the partition λ.
Theorem 3.1. [Fou72, Theorem 1] Suppose λ n 1 and r Z/n.
⊢ ≥ ∈
Then
1
(8) chχr Sn= c (r)p .
↑Cn n ℓ (ℓn/ℓ)
ℓ|n
X
The following straightforward result, essentially implicit in [Sta99,
7.88(a), p. 541], connects and generalizes 3.1 and 1.6.
8 JOSHUA P.SWANSON
Theorem 3.2. Let H be a subgroup of S and let M be a finite-
n
dimensional H-module with character χM: H C. Then
→
1
(9) chM Sn= c p
↑H H µ µ
| | µ⊢n
X
and, for all λ n,
⊢
1
(10) M Sn,Sλ = c χλ(µ),
h ↑H i H µ
| | µ⊢n
X
where
c := χM(h)
µ
h∈H
X
τ(h)=µ
and τ(σ) denotes the cycle type of the permutation σ.
Proof. Write N := M Sn. By definition (see [Sta99, p. 351]),
↑H
χN(µ)
(11) chN = p
µ
z
µ
µ⊢n
X
where z is the order of the stabilizer of any permutation of cycle type
µ
µ under conjugation. From the induced character formula (see [Ser77,
7.2, Prop. 20]), we have
1
χN(σ) = χM(aσa−1).
H
| | s.t. aaXσ∈aS−n1∈H
Say τ(σ) = µ. Each aσa−1 = h H with τ(h) = µ appears in the
∈
preceding sum z times, since σ and h are conjugate and z is also the
µ µ
number of ways to conjugate any fixed permutation with cycle type µ
to any other fixed permutation with cycle type µ. Hence
1
(12) χN(µ) = z χM(h).
µ
H
| | h∈H
X
τ(h)=µ
(9) now follows from (11) and (12). (10) follows from (9) in the usual
way using the fact (see [Sta99, (7.76)]) that p = χλ(µ)s . (cid:3)
µ λ λ
Note that (9) specializes to 3.1 and (10) specPializes to 1.6 when
M = χr. In that case, the only possibly non-zero c arise from µ =
µ
(ℓn/ℓ) for ℓ n.
|
TABLEAUX WITH MODULAR MAJOR INDEX 9
4. Proof of the Main Result
We now turn to the proof of 1.5 and 1.9. We begin by giving
a sufficient condition in terms of upper bounds on symmetric group
character ratios, 4.1, which in turn reduces to a sufficient condition in
terms of lower bounds on fλ, 4.2. We then give an inequality between
hook length products and “opposite” hook length products, 4.4, from
which we classify λ for which fλ < n3. As we will see, 1.5 follows in
almost all cases, with the remainder being handled by brute force and
case-by-case analysis. 1.9 will be similar, except the bound fλ < n5
will be used.
Lemma 4.1. Pick λ n and d R. Suppose for all 1 = ℓ n where λ
⊢ ∈ 6 |
may be written as s := n/ℓ successive ribbons each of length ℓ that
χλ(ℓs) 1
(13) | | .
fλ ≤ ndφ(ℓ)
Then for all r Z/n,
∈
a 1 1
λ,r
< .
fλ − n nd
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. By 1.6, we must sh(cid:12)ow (cid:12)
(cid:12) (cid:12)
1 χλ(ℓs) 1
(cid:12) c (r)(cid:12) < .
n (cid:12) fλ ℓ (cid:12) nd
(cid:12) ℓ|n (cid:12)
(cid:12)X (cid:12)
ℓ6=1
(cid:12) (cid:12)
(cid:12) (cid:12)
Using the explicit form f(cid:12)or cℓ(r) in 1.6 (cid:12)and the fact that n has fewer
(cid:12) (cid:12)
than n proper divisors, it suffices to show
χλ(ℓs) 1
φ(ℓ)
fλ ≤ nd
(cid:12) (cid:12)
(cid:12) (cid:12)
for all ℓ n, ℓ = 1, so the r(cid:12)esult follow(cid:12)s from our assumption (13). (cid:3)
| 6 (cid:12) (cid:12)
Corollary 4.2. Let λ n. If fλ n3 1, then a = 0.
λ,r
⊢ ≥ ≥ 6
Proof. WeapplythefollowingversionofStirling’sapproximation[Spe14,
(1.53)]. For all m Z ,
>0
∈
1 1 1
m+ lnm m+ln√2π lnm! m+ lnm m+ln√2π+ .
2 − ≤ ≤ 2 − 12m
(cid:18) (cid:19) (cid:18) (cid:19)
Using the Fomin-Lulov bound in 1.7, we have
χλ(ℓs) n!ℓn/ℓ
| | ℓ
fλ ≤ (n!)1/ℓ(fλ)1−1/ℓ
10 JOSHUA P.SWANSON
so that
(14)
χλ(ℓs) n n 1 1
ln | | ln !+ lnℓ lnn! 1 lnfλ
fλ ≤ ℓ ℓ − ℓ − − ℓ
(cid:18) (cid:19)
(cid:16) (cid:17)
n 1 n n ℓ n
+ ln +ln√2π+ + lnℓ
≤ ℓ 2 ℓ − ℓ 12n ℓ
(cid:18) (cid:19)
1 1 1
n+ lnn n+ln√2π 3 1 lnn
− ℓ 2 − − − ℓ
(cid:18)(cid:18) (cid:19) (cid:19) (cid:18) (cid:19)
ℓ 1 1 5 1
= + 1 ln√2π lnℓ 1 lnn.
12n − ℓ − 2 − 2 − ℓ
(cid:18) (cid:19) (cid:18) (cid:19)
At ℓ = 2, the right-hand side of (14) is less than ln 1 for n 3.
φ(2)n ≥
At ℓ = 3,4,5, the same expression is less than ln 1 for n 4,3,5,
φ(ℓ)n ≥
respectively. At ℓ 6, we have
≥
χλ(ℓs) 1 1 5 1
(15) ln | | +ln√2π ln6 1 lnn
fλ ≤ 12 − 2 − 2 − 6
(cid:18) (cid:19)
which is less than ln 1 for n 4. Thus, 4.1 applies with d = 1 for all
n2 ≥
n 5, so that
≥
a 1 1
λ,r
< ,
fλ − n n
(cid:12) (cid:12)
(cid:12) (cid:12)
and in particular a = 0.(cid:12) One ma(cid:12)y easily check that fλ < n3 for
λ,r
6 (cid:12) (cid:12)
1 < n 4, and a = 1 = 0. (cid:3)
(1),0
≤ 6
We next give techniques that are well-adapted to classifying λ n
⊢
for which fλ < nd for fixed d. We begin with a curious observation,
4.4, which we have not been able to locate in the literature (though
contrast it with [FL95, Theorem 2.3]).
Definition 4.3. Consider a partition λ = (λ ,...,λ ) with λ λ
1 m 1 2
≥ ≥
0 as a set of cells (in French notation)
··· ≥
λ = (a,b) Z Z : 1 b m,1 a λ .
b
{ ∈ × ≤ ≤ ≤ ≤ }
Given a cell c = (a,b) λ N N, the opposite hook length hop at c
∈ ⊂ × c
is a+b 1. For instance, the unique cell in λ = (1) has opposite hook
−
length 1, and the opposite hook length increases by 1 for each north or
east step.
It is easy to see that hop = h . On the other hand, we
c∈λ c c∈λ c
have the following inequality for their products.
P P