1. Introduction
In the prequel [
1] of this paper we defined a representation of the Hecke algebra of type
A on spaces of superpolynomials. By using the theory of vector-valued nonsymmetric Macdonald polynomials developed by Luque and the author [
2] we constructed nonsymmetric Macdonald superpolynomials. The basic theory including Cherednik operators, the Yang–Baxter graph method for computing the Macdonald superpolynomials, and norm formulas were described. The norm refers to an inner product with respect to which the generators of the Hecke algebra are self-adjoint. The theory relies on relating the Young tableaux approach to irreducible Hecke algebra modules to polynomials in anti-commuting variables. Furthermore, that paper showed how to produce symmetric and anti-symmetric Macdonald superpolynomials, and their norms, by use of the technique of Baker and Forrester [
3]. In the present paper, we consider the evaluation of the polynomials at certain special points. The class of polynomials which lead to attractive formulas in pure product form is relatively small. These values are expressed by shifted
q-factorials, both ordinary (positive integer labeled) and the type labeled by partitions, and
-hook products.
In
Section 2, one finds the necessary background on the Hecke algebra of type
A and its representations on polynomials in anti-commuting (
fermionic) variables and on superpolynomials which combine commuting (
bosonic) and anti-commuting variables. This section also defines the Cherednik operators, a pairwise commuting set, whose simultaneous eigenvectors are called nonsymmetric Macdonald superpolynomials. They are constructed starting from degree zero by means of the Yang–Baxter graph. The necessary details from [
1] are briefly given.
Section 3 presents the main results with proofs about the evaluations; there are two types with similar arguments. The methods rely on steps in the graph to determine the values starting from degree zero. Some of the arguments are fairly technical computations using products of generators of the Hecke algebra. The definition of
-hook products and their use in the evaluation formulas are presented in
Section 4. The evaluations are extended to Macdonald polynomials, of the types studied in the previous sections, with restricted symmetry and antisymmetry properties in
Section 4. The conclusion and ideas for further investigations in
Section 6 conclude the paper.
2. Background
2.1. The Hecke Algebra
The Hecke algebra
of type
with parameter
t is the associative algebra over an extension field of
, generated by
subject to the braid relations
and the quadratic relations
where
t is a generic parameter (this means
for
, and
). The quadratic relation implies
. There is a commutative set of
Jucys–Murphy elements in
defined by
for
, that is,
Simultaneous eigenvectors of form bases of irreducible representations of the algebra. The symmetric group is the group of permutations of and is generated by the simple reflections (adjacent transpositions) , where interchanges and fixes the other points (the satisfy the braid relations and ).
2.2. Fermionic Polynomials
Consider polynomials in
N anti-commuting (fermionic) variables
. They satisfy
and
for
. The basis for these polynomials consists of monomials labeled by subsets of
:
The polynomials have coefficients in an extension field of with transcendental , or generic satisfying for and .
Definition 1. and for . The fermionic degree of is .
This is a brief description of the action of
on
: suppose
implies
, and
implies
: then
Then satisfy the braid and quadratic relations.
There are two degree-changing linear maps which commute with the Hecke algebra action.
Definition 2. {{For set and}} for , set . Define the operators and by and for , while for (also implies and ). Define and .
It is clear that . For let .
Proposition 1. M and D commute with for and
The spaces
and
are irreducible
-modules and are isomorphic under the map
and are of isotype
.The representations of
occurring in this paper correspond to reverse standard Young tableaus (RSYT) of hook shape (see Dipper and James [
4] for details of the representation theory) These are labeled by partitions
of
N and are graphically described by Ferrers diagrams: boxes at
. The numbers
are entered in the boxes in decreasing order in the row and in the column. For a given RSYT
Y let
be the entry at
and define the content
. The vector
is called the content vector of
Y. It defines
Y uniquely (trivially true for hook tableaux). The representation of
is defined on the span of the RSYT’s of shape
in such a way that
for
. We use a space-saving way of displaying an RSYT in two rows, with the second row consisting of the entries
.. Note that
always.
As example let
and
.
We showed [
1] that
is a direct sum of the
-modules corresponding to
and
;
and
, respectively.
2.3. The Module
The basis of
is described as follows: Let
and for
let
. Associate
E to the RSYT
which contains the elements of
E in decreasing order in column 1, that is,
, and the elements of
in
. In the example
with
. The content vector of
E is defined by
. For each
there is a polynomial
such that
for
, and if
then
. In particular if
then
(and this is one of the two cases that are used here). For example, suppose
then
, and .
2.4. The Module
The basis of
is described as follows: Let
and for
let
. Associate
F to the RSYT
which contains the elements of
F in decreasing order in column 1, that is,
, and the elements of
in
. In the example (
4)
with
. As before the content vector of
F is defined by
. For each
there is a polynomial
such that
for
, and if
then
. Note that
implies
and the maximum value occurs at
. This case is the second of those to be studied here. For this set
. As example let
then
, and .
2.5. Superpolynomials
We extend the polynomials in
by adjoining
N commuting variables
(that is
for all
). Each polynomial is a sum of monomials
where
and
. The partitions in
are denoted by
(
if and only if
). The
fermionic degree of this monomial is
and the
bosonic degree is
. The symmetric group
acts on the variables by
and on exponents by
for
(consider
x as a row vector,
as a column vector and
w as a permutation matrix,
, then
and
). Thus,
. Let
. Then using the decomposition
let
The Hecke algebra
is represented on
. This allows us to apply the theory of nonsymmetric Macdonald polynomials taking values in
-modules (see [
2]).
Definition 3. Suppose and then set Note that
acts on the
variables according to Formula (
3).
Definition 4. Let and for and The operators
are Cherednik operators, defined by Baker and Forrester [
5] (see Braverman et al. [
6] for the significance of these operators in double affine Hecke algebras). They mutually commute (the proof in the vector-valued situation is in [
2] [Thm. 3.8]). The simultaneous eigenfunctions are called nonsymmetric Macdonald polynomials. They have a triangularity property with respect to the partial order ⊳ on the compositions
, which is derived from the dominance order:
The rank function on compositions is involved in the formula for an NSMP.
Definition 5. For then and (that is, ). A consequence is that , the nonincreasing rearrangement of , for any , and if and only if .
Theorem 1 ([
2] (Thm. 4.12)).
Suppose and , then there exists a -simultaneous eigenfunctionwhere and its coefficients are rational functions of . Furthermore, where for The exponents and .
The applications in the present paper require formulas for the transformation (called a
step)
when
:
and for the
affine step:
Two other key relations are implies and implies .
3. Evaluations and Steps
We consider two types of evaluations: (0) , , with for , and ; (1) , , with for , and .
Definition 6. Let , . Let , .
Conceptually the two derivations are very much alike, but there are differences involving signs and powers of t that need careful attention. We begin by expressing and in terms of and . Since we are concerned with evaluations the following is used throughout:
Definition 7. For a fixed point and let . In particular if then let for . If then and .
In terms of
b the evaluation formula for
is
The following are used repeatedly in the sequel.
Lemma 1. Suppose for some there is a polynomial and a point y such that and then .
Proof. By hypothesis and thus . Then . □
Lemma 2. Suppose for some there is a polynomial and a point y such that and then .
Proof. By hypothesis and thus . Then . □
In type (0) for which implies for .
Lemma 3. Suppose is of type (0) and for then for some constant depending on x, and for .
Proof. From and Lemma 1 it follows that for . Thus, for , and this implies is a multiple of (the contents determine uniquely). Furthermore for (since are in the same row of ). □
Proposition 2. Suppose and (implying and then Proof. From (
7) and (
8) with
it follows that
because
satisfies the hypotheses of the Lemma implying
. □
The following products are used to relate to .
Definition 8. Let . Suppose and and then Note that the argument of
is
and there are
factors, where
Lemma 4. If then .
Proof. The only factor that appears in but not in is . □
For the special case type (0)
we find
and
Proposition 3. Suppose then and Proof. By Lemma 3
is a multiple of
. For the product formula argue by induction on
. If
then
. If
then
□
In type (1) for which implies for .
Lemma 5. Suppose is of type (1) and for then for some constant depending on x, and for .
Proof. From and Lemma 2 it follows that for . Thus, for , and this implies is a multiple of (the contents determine uniquely). Thus, for (since are in the same column of ). □
Proposition 4. Suppose and (so that and z = then Proof. From (
7) and (
8) with
it follows that
because
satisfies the hypotheses of the Lemma implying
. □
Proposition 5. Suppose then and Proof. By Lemma 5
is a multiple of
. For the product formula argue by induction on
. If
then
. If
then
□
We will use induction on the last nonzero part of
to derive
. Suppose
and
for
where
in type (0) and
in type (1). Define compositions in
by
where
in type (0) and
in type (1). The transitions from
and from
use Propositions 3 and 5. The affine step
and the steps
require technical computations.
Proposition 6. Suppose and are given by (9) then Proof. The spectral vector of has for while and for . The product is □
Proposition 7. Suppose and δ is as in (10) then Proof. The relevant part of
is
for
and
. Thus
and this product telescopes. □
Proposition 8. Suppose and are given by (9) then Proof. The spectral vector of
has
for
while
and
for
. Furthermore,
. Then
Combine this with . □
Proposition 9. Suppose and δ is as in (10) then Proof. The relevant part of
is
for
and
. Thus
and this product telescopes to
. The use of
completes the proof. □
The methods used in these calculations are similar to those used in [
7] for evaluations of scalar valued Macdonald polynomials, however the following computations (from
to
) are significantly different.
Each of the remaining transitions is calculated in its own subsection. The following two lemmas will be used in both types. Recall for any i.
Lemma 6. Suppose and then Proof. From
we get
thus
□
The next formula is a modified braid relation.
Lemma 7. Suppose or then Proof. Expand
which is symmetric in
since
. If
and
then
. □
3.1. From to for Type (0)
In this section, we will prove
. Start with
(where
for
and
otherwise). Let
and
for
(so that
in (
9)). Abbreviate
. If
then
. Set
for
, then
To start set
and
(thus
)
Two series of points are used in the calculation: Define
,
,
for
; define
for
. Thus
Lemma 8. Suppose and then = and for .
Proof. This follows from
with
so that
, and with
. If
then
. □
Proof. The formula is true for
by (
12). Assume it holds for some
i then
and
Then
satisfies the hypotheses of Lemma 8 for
and
The first part of the right side of (
14) combined with (
16) gives
which cancels out the first part of the right side of (15) combined with (
18); note the factor
. The terms that remain are exactly the claimed formula for
. □
Proposition 11. .
Proof. Set
in (
13). To complete the proof we need to show
By construction
and
for
. This implies
and
for
. Let
then
for
This property defines
up to a multiplicative constant, and thus
(because
satisfies
for
and thus
). To set up an inductive argument let
and set
for
. Then
if
or
,
and
. Claim that
The first step is
. Note
for
. Suppose the formula is true for some
, then
This proves the formula. Set
then
. By definition
and so
. Now
. Thus
because
. □
Next we consider the transition from
to
(see (
9)) with the affine step
(recall
). To get around the problem of evaluation at the
q-shifted point we use
thus
where
. From the previous formula we see that we need to evaluate the right hand side at
and apply
. Since
for
it follows that
for
.
Definition 9. Let and for .
The corresponding evaluation formula is
Proposition 12. Suppose then
Proof. From
for
it follows that
if
. By (
20)
. Suppose
then
satisfies
for
so that
and
,
is a multiple of
and
. Thus,
and this holds for
. □
Recall the points
given by
for
. Define
for
. By the braid relations
These products are used in the proofs:
If then commutes with .
Lemma 9. Suppose for and for then Proof. Let
then
and
. Thus
Repeated application of this relation shows = . □
If then and .
Proof. Proceed by induction. By (
11)
and
. Thus, the formula is valid for
(with
). Suppose it holds for some
, then
and
Combine with formula (
22) to obtain
For the part in (23)
thus
The first part leads to
because
and
. Then
by Lemma 9, so combine to obtain
which cancels the second term in (
24). The second part gives
by Lemma 7, formula (
21) and
. □
Proposition 14. .
Proof. Set
in (
22) thus
By Lemma 9
. Furthermore
and thus
and by Lemma 1
for
. This implies
. However,
and this is proved by an argument like the one used in Proposition 11. Let
and set
for
. Claim
Suppose the formula is true for some
then
and this is the formula for
. Then
and
, and thus
and
. As in Proposition 11 this implies
, and this completes the proof. □
3.2. Evaluation Formula for Type (0)
Recall the intermediate steps:
Proposition 15. Suppose satisfies and for with then where for and .
Proof. The leading factors are . □
Corollary 1. Suppose λ is as in the Proposition and satisfies for and then Proof. This uses
with
. □
This formula can now be multiplied out over k, starting with , where .
Theorem 2. where and .
Proof. For
define
by
for
and
for
. Formula (
25) gives the value of
. For fixed
the products
contribute
to
(the product telescopes). Each pair
with
contributes
. If
then
and thus
k can be replaced by
in the above formulas. The exponents on
follow easily from
. □
Remark 1. Recall the leading term of , namely , where . By using for one finds that so that and .
There is a generalized
-Pochhammer symbol
and the
k-product in (
27) can be written as
. In a later section we will use a hook product formulation which incorporates a formula for
.
3.3. From to for Type (1)
To adapt the results for type (0) to type (1) it almost suffices to interchange and replace t by . However, there are signs and powers of t, and different formulas involving to worry about. The interchange occurs often enough to get a symbol:
Definition 10. Suppose is a function of (possibly also depending on λ or α) then set
We will reuse some notations involving
and so forth, with modified definitions (but conceptually the same). In this section, we will prove
=
. Start with
(where
for
and
otherwise). Let
and
for
(so that
in (
9)). Abbreviate
. If
then
. Set
for
, then
These are analogs of the type (0) definitions, with
:
Recall for . (The proof of the following is mostly the same as that for Proposition 10 except for signs and powers of t.)
Proof. The transformation from
to
is in (
28). Specialize to
and
so that
,
, and
The values from (
29) are
,
. Thus
then
From the spectral vector of
it follows that
for
and
. Thus,
=
and
Then
appears in the expression for
with factor
and
with factor
and the two cancel out (
). This proves the inductive step. □
Proposition 17. .
Proof. Set
in (
30). Claim
. From
and
for
it follows that
for
. This implies
for some constant (similarly to the argument in Proposition 11
satisfies
for
implying
). Let
and
then define
for
. Use induction to show
Assume the formula is true for some
then
(because ). Thus, and □
Next we consider the transition from
to
(see (
9)) with the affine step
and as before the calculation is based on the formula
where
. From the previous formula we see that we need to evaluate
. Since
for
it follows that
for
.
Definition 11. Let and for .
Proposition 18. Suppose then
Proof. From
for
it follows that
if
. By (
20)
. Suppose
then
satisfies
for
so that
and
,
is a multiple of
and
. Thus,
and this holds for
. □
Similarly to the type (0) computations let
Lemma 10. Suppose for and for then Proof. Let
then
and
. Thus
Repeated application of this formula shows
□
Proof. Proceed by induction. By (
11)
and
. Thus, the formula is valid for
(with
). Suppose it holds for some
, then
and
Combine with formula (
31) to obtain
For the second line (32)
thus
The first part leads to
because
and
. Then
by Lemma 10, so combine to obtain
which cancels the second term in (
33). The second part gives (using the braid relations in (7))
by (
21) and
□
Proposition 20. .
Proof. Set
in (
31)
and
(note
) thus
Now
thus
satisfies the hypothesis of Lemma 10 with
and
Since
and
for
it follows that
for the same
i values and hence
(with
because
lie in the same row of
). Take
and
then
because
. Continue this process to obtain
thus
because
. Thus,
=
. □
3.4. Evaluation Formula for Type (1)
Recall the intermediate steps:
Proposition 21. Suppose satisfies and for with then where for and .
Proof. The leading factors are , since and . □
Corollary 2. Suppose λ is as in the Proposition and satisfies for and then Proof. This uses formula (
26). □
This formula can now be multiplied out over k, starting with , where .
Theorem 3. where and .
Proof. This is the same argument used in Theorem 2 by the application of . □
Remark 2. Recall the leading term of , namely , where . By using for one finds that so that and .
4. Hook Product Formulation
Recall the definition of the
-hook product
where
and
, where the length of
is
. The terminology refers to the Ferrers diagram of
which consists of boxes at
.
Proposition 22. Suppose and for some fixed then Proof. The argument is by implicit induction on the last box to be added to the Ferrers diagram of
. Suppose
for
and
. Define
by
for all
i except
. Denote the product on the left side of (
35) by
, then
the
j-product telescopes. Adjoining a box at
to the diagram of
causes these changes:
for
,
for
. The calculation also uses
;
. Thus
because the change in the product for row
is
Denote the second product in (
35) by
then
Hence To start the induction let , then , while and . This completes the proof. □
Note that (the generalized -Pochhammer symbol). Setting in the Proposition leads to another formulation:
The same method can be applied to by using (Definition 10).
There is a modified definition of leg-length for arbitrary compositions
:
Suppose
then
from [
2] [p.15,Prop. 5] (the argument relates to the box at
in the Ferrers diagram of
and the change in its leg-length) so that
Suppose
then from
(see Proposition (3)) and (
27) we obtain
There is a slight complication for type (1)
Start with
for
(because
) and then
. Suppose
and apply
in (
36) to obtain
combine with
and then
Thus,
, and
We have shown that the values of certain Macdonald superpolynomials at special points or are products of linear factors of the form where and .
5. Restricted Symmetrization and Antisymmetrization
A type of symmetric Macdonald superpolynomial has been investigated by Blondeau et al. [
8]. The operators used in their work to define Macdonald polynomials are significantly different from ours. There are results on evaluations for these polynomials found by González and Lapointe [
9]. In this section, we consider symmetrization over a subset of the coordinates, and associated evaluations.
Fix and consider the sum satisfying for . In this section, we determine . Similarly fix and consider the sum satisfying for , then evaluate .
Lemma 11. Suppose and for some i. Let and let . If then and if then
Proof. The general transformation rules are given in matrix form with respect to the basis
One directly verifies that,
□
Definition 12. For set .
Proposition 23. Suppose then satisfies for .
Proof. Fix
i. If
and
then
because
and thus
. Otherwise take
and
then set
by Lemma 4, and
. By Lemma 11
. For each
i the sum for
splits into singletons (
and pairs
. Each piece is annihilated by
. □
There is now enough information on hand to find
, since
This sum can be evaluated using the norm formula established in [
1]. This formula applies to arbitrary
and arbitrary sets
, In the present context which uses only
with
the formula is used with
N replaced by
and the reverse
of
is replaced by
For
define
. For
and
let
. The formula from [
1] specializes to
Note that the multiplier is a type of
t-multinomial symbol. It is straightforward to show
This product can be combined with the
-product in (
27) to show:
Definition 13. For let =
Lemma 3 applies to each in the sum for thus for .
Proposition 24. is symmetric in y. In particular for any permutation u of (that is ).
Proof. Suppose
then
The latter is a polynomial identity (after multiplying by ) and thus holds for all , and hence . □
Next we consider asymmetric polynomials in type (1). Recall implies if and if .
Definition 14. For set .
Proposition 25. Suppose then satisfies for .
Proof. Fix
i. If
and
then
because
and thus
. Otherwise take
and
then set
by Lemma 4, and
. By Lemma 11
. For each
i the sum for
splits into singletons (
and pairs
. Each piece is annihilated by
. □
Similarly to the symmetric case we can determine
since (by Proposition 5)
Formula (
37) can be adapted to find the sum by applying
and chasing powers of
t (in
for example). The typical term in
is
and applying
yields
with
, the typical term in
(after the interchange
). Thus
From
and
it follows that
Now let
and consider
and the transformed
We find
and we have shown
Similarly to type (0) this formula can be further developed:
Definition 15. For let =
Lemma 5 applies to each in the sum for thus for .
Proposition 26. is symmetric in y. In particular for any permutation u of (that is ).
Proof. The latter is a polynomial identity (after multiplying by ) and thus holds for all , and hence . □
6. Conclusions and Future Directions
In the context of Macdonald polynomials, “evaluation” refers to finding a closed form consisting of a product of linear factors for the value of a polynomial at a certain point. The polynomials are sums of monomials whose coefficients are rational functions of
. The linear factors are of the form
where
and
. Any ordinary (scalar-valued) nonsymmetric Macdonald polynomial does have an evaluation formula at the point
(see [
7] [Prop. 5]) (this is a multi-variable analog of the value of a Gegenbauer polynomial
at
). However, in the vector-valued case, computational experiments suggest that there are no generally applicable formulas of this type. In the present paper we established evaluations of a relatively restricted class of nonsymmetric polynomials at special points. The labels
of the Macdonald polynomials
have only two possibilities out of many,
for the isotype
.
There are other possible evaluations that deserve to be investigated: these relate to singular polynomials. This refers to the situation where the parameters
satisfy a relation like
and a polynomial
satisfies
for
The Jucys–Murphy operators on
are defined in terms of
(see (
5)):
for
. Of course, finding these singular parameters
is already a research problem by itself. For small
N and degree we can find some examples (with computer algebra) and test evaluations. It appears there are interesting results to find.
Consider
and
(of isotype
). The spectral vector of
is
. Let
The polynomial
is singular for
and
Similarly
is singular at
; its spectral vector is
and for
As well
is singular at
; its spectral vector is
and for
For an example with higher degree consider
and
(the isotype is
) Then
and
is singular for
At
we find
We would expect an evaluation formula involving the elements of the spectral vector for which
and with as many free variables as nonzero elements of
. There is a nice necessary condition for a singular value: the
t-exponents of the specialized spectral vector have to agree with the content vector of an RSYT. For example, set
in
with the result
, and
is the content vector of
Then with can not be singular at : the spectral vector and is not the content vector of any RSYT.
There are interesting results dealing with singular Macdonald superpolynomials waiting to be found.