1. Introduction
In the study of polynomials in several variables there are two approaches, one is algebraic, which may involve symmetry groups generated by permutations of coordinates and sign changes, for example, and the analytic approach, which includes orthogonality with respect to weight functions and related calculus. The two concepts are combined in the theory of Dunkl operators, which form a commutative algebra of differential-difference operators, determined by a reflection group
G and a parameter, and which are an analog of partial derivatives. The relevant weight functions are products of powers of linear functions vanishing on the mirrors and which are invariant under the reflection group
G. In the particular case of the objects of our study, namely the symmetric groups
, an orthogonal basis of polynomials (called
nonsymmetric Jack polynomials) is constructed as the set of simultaneous eigenfunctions of the Cherednik–Dunkl operators. This is a commutative set of operators, self-adjoint for an inner product related to the weight function. The inner product is positive-definite for an interval of parameter values but for a discrete set of values there exist null polynomials (that is,
). It is these parameter values that concern us here. The set of such polynomials of minimal degree has an interesting algebraic structure: in general it is a linear space and an irreducible module of
. The theory for scalar polynomials is by now well understood [
1], and the open problems concern
vector-valued polynomials whose values lie in irreducible modules. That is, the symmetric group
acts not only on the domain but also the range of the polynomials. The key device for dealing with the representation theory is to analyze when a polynomial is a simultaneous eigenfunction of the Cherednik–Dunkl operators and of the Jucys-Murphy elements with the same respective eigenvalues. In Etingof and Stoica [
2] there is an analysis of the vanishing properties, that is, the zero sets, of singular polynomials of the groups
as well as results on singular polynomials associated with minimal values of the parameter for general modules of
and for the exterior powers of the reflection representation of any finite reflection group
G (see also [
3]). Their methods do not involve Jack polynomials. Feigin and Silantyev [
4] found explicit formulas for all singular polynomials which span a module isomorphic to the reflection representation of
G.
This paper concerns polynomials taking values in the representation of the symmetric group corresponding to a rectangular partition. In particular for (the superscript indicates multiplicity) we construct nonsymmetric Jack polynomials in variables which are singular (annihilated by the Dunkl operators) for the parameter and which span a module isomorphic to the representation .
In
Section 2 we present the basic definitions of operators, combinatorial objects used in the representation theory of the symmetric groups, and vector-valued nonsymmetric Jack polynomials.
Section 2.1 is a concise treatment of the formulas for the transformations of the Jack polynomials under the simple reflections; in a sense the whole paper is about the effect of various transformations on these polynomials. Our combinatorial arguments depend on
bricks, our term for the
rectangles making up the partitions of concern; the properties of bricks and the tableaux built out of them are covered in
Section 3. The Jucys–Murphy elements form a commutative subalgebra of the group algebra of
and are a key part of the proof that certain Jack polynomials are singular. The details are in
Section 4. To show singularity we establish the existence of the needed Jack polynomials when the parameter is specialized to
and the machinery for this is developed in
Section 5 and
Section 6. The existence of a class of singular polynomials leads to constructing maps of modules of the rational Cherednik algebra; this topic is covered in
Section 7. Finally
Section 8 discusses an easy generalization and also describes an example which demonstrates the limits of the theory and introduces open problems.
2. Background
The symmetric group acts on by permutation of coordinates. The space of polynomials is where is a parameter and . For set . The action of is extended to polynomials by where (consider x as a row vector and w as a permutation matrix, , then ). This is a representation of , i.e., for all . Our structures depend on a transcendental (formal) parameter , which may be specialized to a specific rational value .
Furthermore,
is generated by reflections in the mirrors
for
. These are
transpositions, denoted by
, so that
denotes the result of interchanging
and
. Define the
-action on
so that
that is
(so
is taken as a column vector and
).
The
simple reflections ,
, generate
. They are the key devices for applying inductive methods, and satisfy the
braid relations:
We consider the situation where the group
acts on the range as well as on the domain of the polynomials. We use vector spaces, called
-modules, on which
has an irreducible orthogonal representation. See James and Kerber [
5] for representation theory, including a discussion of Young’s methods.
Denote the set of
partitionsAn irreducible representation
of
corresponds to a partition of
N given the same label, that is
and
. The length of
is
. There is a Ferrers diagram of shape
(also given the same label), with boxes at points
with
and
. A
tableau of shape
is a filling of the boxes with numbers, and a
reverse standard Young tableau (RSYT) is a filling with the numbers
so that the entries decrease in each row and each column. The set of RSYT of shape
is denoted by
and the representation is realized on
. For
and
the entry
i is at coordinates
and the
content is
. We use
to denote the entry at
, so
. Each
is uniquely determined by its
content vector . A sketch of the construction of
is given in
Section 2.1.3 and Remark 1. We are concerned with
, i.e., the
which is equipped with the
action:
extended by linearity to
Definition 1. The Dunkl and Cherednik–Dunkl operators are () extended by linearity to all of .
The commutation relations analogous to the scalar case hold, i.e.,
The simultaneous eigenfunctions of
are called (vector-valued) nonsymmetric Jack polynomials (NSJP). They are the type
A special case of the polynomials constructed by Griffeth [
6] for the complex reflection groups
. For generic
these eigenfunctions form a basis of
(
generic means that
where
and
). They have a triangularity property with respect to the partial order ⊳ on compositions, which is derived from the dominance order:
There is a subtlety in the leading terms, which relies on the
rank function
:
Definition 2. For then
A consequence is that , the nonincreasing rearrangement of , for any . For example if then and (recall ). Also if and only if .
For each
and
there is a NSJP
with leading term
, i.e.,
where
; the coefficients are rational functions of
. These polynomials satisfy
For detailed proofs see [
7]. The commutation
implies
This introduces the definition of
Jucys–Murphy elements in the group algebra
which satisfy
They act on
by
We will use the modified operators
. The associated spectral vector is
so that
for
.
Throughout we use the phrase “at ” where is a rational number to mean that the operators and polynomials are evaluated at . The transformation formulas and eigenvalue properties are polynomial in x and rational in . Thus, the various relations hold provided there is no pole. Hence to validly specialize to it is necessary to prove the absence of poles.
Suppose
and
then
if and only if
at
(obvious from (
2)). The polynomial
p is said to be
singular and
is a
singular value. From the representation theory of
it is known that an irreducible
-module is isomorphic to an abstract space whose basis consists of RSYT’s of shape
, a partition of
N. The eigenvalues of
form content vectors which uniquely define an RSYT. Suppose
is a partition of
N then a basis
(of an
-invariant subspace) is called
a basis of isotype if each
for
and each
. If some
is a simultaneous eigenfunction of
with
for
then the representation theory of
implies that
is the content vector of a uniquely determined RSYT of shape
for some partition
of
N; this allows specifying the isotype of a single polynomial without referring to a basis. The key point here is when a subspace does have a basis of isotype
made up of NSJP’s. specialized to a fixed rational
.
In this paper, we construct singular polynomials for the partition of for the singular value and which are of isotype , with . To show that the nonsymmetric Jack polynomials in the construction have no poles at we use the devices for proving uniqueness of spectral vectors and performing valid transformations of the polynomials. The proof of singularity will follow once we show the relevant polynomials are eigenfunctions of the Jucys–Murphy operators . These properties are proven by a sort of induction using the simple reflections . For this purpose we describe the effect of on .
One key device is to consider the related tableaux as a union of k rectangles of shape , which we call bricks.
2.1. Review of Transformation Formulas
We collect formulas for the action of
on
. They will be expressed in terms of the spectral vector
and (for
)
The formulas are consequences of the commutation relationships: for and ; for . Observe that the formulas manifest the equation .
2.1.1. Case:
2.1.2. Case:
2.1.3. Case:
In this case . Then if
()
,
(
)
Remark 1. The previous four formulas when restricted to (so that and ) describe the action of τ on ; in this situation for .
There is an important implication when and (at ) the general relation becomes provided that does not have a pole. Our device for proving this is to show uniqueness of the spectral vector of or of another polynomial which can be transformed to by a sequence of invertible () steps using the simple reflections .
3. Properties of Bricks
A brick is a
tableau which is one of the
k congruent rectangles making up the Ferrers diagram of
or
. Since it is clear from the context we can use the same name for the appearance in
or
. Let
, then
is the part
of
or the part
in
. The
standard brick has the entries entered column by column:
In this section, we use bricks to construct for each a pair such that for . Later on we will prove uniqueness of .
For the partition
we use the distinguished RSYT
formed by entering
column by column, i.e.,
is the concatenation of
. Observe
, a Catalan number. The contents of
in
are given by
Form the distinguished RSYT
of shape
by stacking the standard bricks, from
at the top (rows
and
) to
at the bottom (rows
and
). The location of
in
has corners
,
,
,
(and the entries are
entered column by column). Thus,
has the numbers
entered column by column in each brick; here is the example
The contents for
in
are
Let
.
Proposition 1. The spectral vector at of equals the content vector of .
Proof. Since if then and . By the structure of it suffices to check the value at one corner, say the top left one. Taking with we obtain . ☐
Suppose
then there is a permutation
of
and an RSYT of shape
such that
at
for
. The construction is described in the following.
Definition 3. Suppose and ; and suppose the part of S in the brick is The entries decrease in each column and in each row. Define by for . Set up a local rank function (for ): then and . Picturesquely, form a brick-shaped tableau by replacing by and then adding to each entry. Then stack these tableaux to form an RSYT of shape τ. Specifically set , for . Perform this construction for each ℓ with .
Denote
constructed in the Definition by
, or by the abbreviation
.
Proposition 2. Suppose and then for all i.
Proof. Suppose i is in brick and for some j. If then and , while if then and . Then . Also and thus .
If then , and . If then , and . ☐
Here is an example for
with the values of the local rank
now add
respectively and combine to form
For example and thus .
Essentially what is left to do for the singularity proofs is to show
is closed under
and that
for all
i. Here is a small example of the impending difficulty: let
and
What is the result of applying
? Interchanging 5 and 6 in
S results in a tableau violating the condition of decreasing entries in each row (thus outside the span), and the general transformation formula in
Section 2.1.2 (
) says
with
and
, thus
at
. To show that the formula gives
it is necessary to show
has no pole at
. These proofs comprise a large part of the sequel.
4. Action of Jucys–Murphy Elements
The Jucys–Murphy elements satisfy for and for .
Suppose there is a subset with the properties (spectral vectors at , recall ):
and implies for some and ; also ;
and implies ;
implies and thus .
The following is a basic theorem on representations of
and we sketch the proof.
Theorem 1. If satisfies these properties then implies for .
Proof. Arguing by induction suppose
for all
and
. The start
is given in the hypotheses. Let
and suppose that
, then
Next suppose
and set
, thus
(inductive hypothesis). Then
This completes the induction. ☐
We want to show that (as in Definition 3) satisfies the hypotheses of Theorem 1. From the construction it is clear that because and this cell is in . Fix and . Abbreviate . There are several cases:
then and where and for all Specifically if implying that i and are in different bricks then and , while if then and is formed from T by transforming the part of T in interchanging and
(
) then by construction
and
; suppose that
in the notation of Definition 3. By hypothesis
,
. Then
and
This implies
. By
Section 2.1.3 .
and
; then
; using Definition 3
and
for some
j with
or
, and
Thus,
. In the first case
and in the second case
and thus
. by
Section 2.1.3.
and
then
(because the entries of
S are decreasing in each row). Thus,
is in position
or
of
and
i is
or
respectively of
. The relevant transformation formula is in
Section 2.1.2:
. To allow
in this equation and conclude
it is necessary to show
has no poles there.
To complete the proof that for and (at ) we will show each and (as described in (4) above) has no poles at . In the next section, we show that it suffices to analyze specific tableaux.
5. Reduction Theorems
Suppose some was shown to be defined at (no poles) and then where is also defined (recall is a nonzero multiple of if or of if and .) and the process is invertible. In other words if is a valid spectral vector and then is also a valid spectral vector (valid. means that there is a NSJP with that spectral vector and it has no pole at ).
We consider column-strict tableaux
S of shape
which are either RSYT or
S differs by one row-wise transposition from being an RSYT. Their content vectors are used in the argument.
Column-strict means that the entries in each column are decreasing.
Definition 4. Suppose and then is the set of tableaux S of shape σ such that S is column-strict and defined by and for is an RSYT.
Suppose
for
,
and
then
and
is a spectral vector associated with
. Call this a
permissible step. In fact, the inequality
is equivalent to the row and column property just stated. If
then
and if
then
. (because any row or column orderings do not change). For counting permissible steps we define
A permissible step
adds 1 to
. The reduction process aims to apply permissible steps until a inv-maximal tableau is reached. In
the inv-maximal element is
and
.
Definition 5. For and define a distinguished element of by , for and for Then
. Here are two examples with
:
Any can be transformed by a sequence of permissible steps to (this is a basic fact in representation theory but the explanation is useful to motivate the argument for ), and any can be transformed in this way to . For convenience, replace by N since only the number of columns is relevant. Suppose and by permissible steps was transformed to with for (the inductive argument starts with ). From the definition of it follows that and for all This implies and with . Then the list of entries equals Apply in this order (if then already done). Each one is a permissible step, with t in and in with . This produces satisfying for . The induction stops at .
Suppose
and by permissible steps was transformed to
with
,
for
(the inductive argument starts with
). From the definition of
it follows that
and
for all
This implies
and
with
. The numbers
are in
. As in the RSYT case apply
in this order (if
then already done). It is possible that one pair of adjacent entries is out of order (when
) but the argument is still valid. Here is a small example with
.
The inductive process can be continued until and the result is a tableau with the entries in the first columns. Thus, the entries are in the remaining columns.
The next part of the process is to start from the last column and work forward. Suppose
by permissible steps was transformed to
with
for
(the first step is with
). As before
and
for
. This implies
and
with
. The numbers
are in
. This range of cells has contents
,
(excluding
). Possibly one pair of adjacent entries is out of order (when
). In terms of contents while
so the steps
are permissible in that order resulting in
with
. The process stops at
. The result is
for
and
. Thus, the entries in columns
n and
are
. The definition of
forces the position of these entries:
Thus, we showed that any can be transformed by permissible steps to an inv-maximal tableau.
In the above example
and
and the action of
suffices to obtain the desired tableau:
no more permissible steps are possible.
In our applications and with .
6. Uniqueness Theorems
This section starts by showing how uniqueness of spectral vectors is used to prove that specific Jack polynomials exist for some
, that is, there are no poles there.
Proposition 3. Suppose has the property that , and for at implies then is defined at , in the sense that the generic expression for can be specialized to without poles.
Proof. From the ⊳-triangular nature of (
1) it follows that the inversion formulas are also triangular, in particular
where
is a rational function of
. By hypothesis for each
and
there is an index
such that
at
. Recall that the generic spectral vector
uniquely determines
, since
is found from the coefficients of
and the remaining terms of
determine the content vector of
. Define an operator on
by
Then annihilates each with and maps to itself. Thus and by construction the right hand side has no poles at . ☐
The condition in the Proposition is sufficient, not necessary. There is an example in the concluding remarks to support this statement.
We introduce a simple tool for the analysis of a pair
, namely the tableau
with the entries being pairs
such that the tableau of just the first entries coincides with
T, i.e., if
then
. As example let
The tableau has order properties: in each row the first entries decrease and the second entries nondecrease (weakly increase), and the same holds for each column.
The first part is to assume and (called the fundamental equation) for and to deduce that and .
Our approach to the uniqueness proofs is to work one brick at a time, and in each brick alternating between even and odd indices showing the values of
and
agree with those of
. For each cell we use the fundamental equation and the order properties of
to set up inequalities which lead to a contradiction if
.
Theorem 2. Suppose , and for then and .
Proof. This is an inductive argument alternating between even and odd indices to prove the desired equalities for brick
. Then the argument is applied to the tableaux with one less brick. Suppose we showed
for
(thus
) and
for
and
for
. The start of the induction is
so the previous conditions are vacuous. Suppose
,
and
. Then
Thus, if
and
then
. Let
with
and
(thus
); furthermore by the inductive hypothesis and
it follows that
. Then
However, and there is a contradiction. Thus, , and (the other possibility for the entry in T is , ruled out by the content value). The start forces . The last step is with and results in .
Suppose we showed
for
and
for
and
for
(the first step is with
). Suppose
,
and
then
If
and
then
and
with
,
(because
is the last appearance of
ℓ in
) and
(thus
). By the inductive hypothesis and
it follows that
. Then
However, and this is a contradiction Thus, and (the other possibilities for the entry in T are and , both ruled out by their content values). The last step of the induction is for .
Replace the original problem by a smaller one: let
the tableau
of shape
with entries
for
and
and the tableaux
and
of shape
with entries
for
and
The consequences of these definitions are with
Then
and the same argument as before shows that
agrees with
in the first four rows (the first two bricks). Repeat this process
times arriving at
for
and the entries of the remaining
are
entered column by column
Thus, and the spectral vector of is unique. ☐
We set up the same argument for removing the last brick
. Intuitively this is already done: rotate the tableaux through
and replace the entry
r by
. This idea guides the proof. The property
implies
for all
i. Here the inductive argument alternates between odd and even indices.
Remark 2. The following is an alternate proof of Theorem 2.
Proof. Suppose we showed
for
,
for
and
for
(the first step is at
with vacuous conditions on
T, the last at
). Suppose
,
and
then (using
)
If
and
then
. Let
with
,
, and
(so that
); furthermore by the inductive hypothesis and
it follows that
, then
However, , a contradiction, thus and which implies (the other possible location for the entry in T is but ). The start is (forced by definition of RSYT) and . The last step results in .
Suppose we showed
for
,
for
and
for
(the first step is at
, the last at
). Suppose
,
and
then
Thus, if
and
then
. Let
with
,
, and
(so that
) then
However, and there is a contradiction. Thus, and ; the other possible locations for the entry in T are , ruled out by their content values.
The inductive process concludes by showing
for
and
for
and
. As before the original problem can be reduced to a smaller one by removing the last brick. This is implemented by defining
as follows:
Clearly and the hypothesis holds for . So the bricks can be removed in the order . At the end there is only one brick all and (for and the corresponding parts (brick ) of T and are identical, and thus to . ☐
The second part is to prove uniqueness for the spectral vectors derived from the content vectors of the tableaux
(see Definition 5) where
, at the edge of brick
adjacent to the edge of
. To prove this we use the previous arguments to remove the bricks above and below bricks
and
s leaving us with a straightforward argument where only two values of
play a part. To obtain the hypothetically unique spectral vectors we apply reflections to
. For brevity let
. First compute
, a nonzero multiple of
; this is a permissible step and hence this polynomial is defined for
. Then form
which produces the polynomial labeled by
whose spectral vector equals the content vector of
. Also form
, with label
associated with
. Here are tables of values of
,
,
in the zone of relevance
;
and the corresponding spectral vectors
, denoted by
for convenience,
The respective cells in
are
,
with respective contents
. Except for these four locations
so in the bricks
for
and
the previous proofs can be applied; the various
and
values apply verbatim.
Theorem 3. Suppose or 2 and such that and for then .
Proof. By the previous arguments we show
for
,
(using the proof for Theorem 2) and
(using the alternate proof following Remark 2 This leaves just two bricks and we can assume
. Reducing
to
results in (with
and
)
The property
implies that
is a permutation of
. The entries
in
T are all in
and the entries
are in
. This follows from
implies
, for if
or 4 then the ordering property of
implies
or
but
or
is impossible (as values of
). Thus, the
pairs
fill
. The next few steps are for
; if
then there are just 4 cells left and the last part of the proof suffices. By using the previous arguments we show
for
and
for
, and also that for
and
and for
As example of the steps of the proof let
then
and let
. Then
However, if
then
(since
) which is impossible; thus
and
Similarly consider
and
then
but if
then
which is impossible; thus
and
.
All but four entries were accounted for and thus
, the rank of the first zero in
, and
the rank of the last 1 in
. Thus,
. The relevant part of the content vector is
The remaining equations are
Let with . Then and . Let with . Then and .
Case: From the table we see that and . This implies and. Thus , with central entries
Case From the table we see that and . This implies and . Thus , with being the central entries.
This concludes the proof. ☐
7. Maps of Standard Modules
The algebra generated by
and multiplication by
for
along with
is the rational Cherednik algebra (type
) and
is called the standard module associated with
, denoted
. In this section, we construct a homomorphism from the module
to
when the parameter
. In the notation of Definition 3 for each
there is a pair
such that the spectral vector
for
at
. However the polynomials
need to be rescaled so that they transform under
w with the same matrix as
. Recall the formula for
which is derived from the requirement that
is an orthogonal basis and each
is an isometry (and we use this requirement for
as well)
By the construction of (column by column) either i is odd and implies and or i is even and implies and , thus .
Suppose
so that
and the transformation rule
Section 2.1.3 yields (with
):
We need two rules for the NSJP :
and
(
Section 2.1.3)
Recall the abbreviation
. The following discussion of
applies only to
at
, which is an irreducible
-module, isomorphic to
. We use the normalization (
)
and this determines the other norms.
Definition 6. For let . By convention .
Proof. We argue by induction on
(see (
3)). Suppose the formula holds for each
with
for some
u; the start is
. Suppose
and
, i.e.,
and
. Then
and
. Also
. For convenience let
. (Recall that
for all
j). There are two cases for the relative locations of
i and
in
S.
If
for some
ℓ then by definition
,
and
are in the same brick
in
T. Then
. By Formula (
4)
Because the product in is invariant under the replacement
If
i and
are in different bricks then
because
. Then
and by Formula (
5)
The product in
is over pairs
with
and
. Changing
S to
leaves the pairs with
alone and interchanges the pairs
and
respectively. The pair
is added to the product since
and thus
This completes the induction. ☐
For
let
denote the matrix of the action of
on the basis
, so that
. These matrices are generated by the
which are specified in the transformation formulas in
Section 2.1.3. The polynomial
is a simultaneous eigenfunction of
with the same respective eigenvalues as
S and it has the same length, thus it satisfies
Note
.
Definition 7. The linear map is given by where each .
Proposition 5. The map μ commutes with multiplication by and with the action of for and .
Proof. The first part is obvious. For the second part let
for some
and
. Then
☐
Recall the key fact:
for
, at
.
Theorem 4. The map μ commutes with for .
Proof. Let
for some
and
then
However,
and thus
. Notice the part
of the calculation vanishes. ☐
The Proposition and the Theorem together show that
is a map of modules of the rational Cherednik algebra (with parameter
). In fact the map can be reversed: define scalar polynomials
by
then it can be shown (fairly straightforwardly) that
is a singular polynomial in
for
and is of isotype
. So one can define a map analogous to
from
. There are general results about duality and maps of modules of the rational Cherednik algebra in ([
8] [Sect. 4]). In [
9] there are theorems about the existence of maps between standard modules in the context of complex reflection groups.
8. Further Developments and Concluding Remarks
The construction of singular polynomials in which are of isotype is easily extendable to with and . Define then is singular for . This is valid because the uniqueness theorems can be derived from the case: suppose such that and , then which implies , for each i. Further and the uniqueness of shows the same for .
It should be possible to extend our analysis to the situation where the top brick is truncated, that is, and with , but we leave this for another time.
In this paper, we constructed singular nonsymmetric Jack polynomials in
which are of isotype
. In general, suppose
and
are partitions of
N and there are singular polynomials in
for
which are of isotype
, then it can be shown that there are singular polynomials in
for
which are of isotype
. This idea was sketched in
Section 7. It turns out that interesting new problems may arise. In the present work, we used uniqueness theorems about spectral vectors to show the validity of specializing NSJP’s to
(some fixed rational) to obtain singular polynomials. However it is possible that some singular polynomial is a simultaneous eigenfunction of
but is not the specialization of an NSJP.
Our example is for
and
. The singular polynomials for
(scalar polynomials) are well-known ([
1]). In particular
has no pole at
and is singular there, furthermore it is of isotype
. From the general result there are singular polynomials in
of isotype
(that is, invariant) for
. The uniqueness approach fails here. Let
The spectral vectors
are (note
)
By direct (symbolic computation assisted) calculation we find that both
and
are defined (no pole) at
, neither is singular or of isotype
(invariant under each
) but
Also is invariant and for . The polynomial is a sum of 100 monomials in x, with coefficients in .
We suspect that our results benefitted from the fact that and are rectangular partitions, and that the analysis of singular polynomials for other partitions (hook tableaux for example) becomes significantly more difficult.