ńņš. 3 |

ā

ā ā ā’1

(2z)2Ī½

ā’1

Ī½+1

E2Ī½ = (ā’1) dz.

ā

ā’ā ā’1 cos Ļz

Slightly diļ¬erent proofs of (3.52) can be found in [1] and [108, App. A.5], together

with more Hankel determinant evaluations (among which are also (3.56) and (3.58),

respectively). The evaluation (3.54) can be derived by considering Charlier polyno-

mials (see [35] for such a derivation in a special case). The evaluation (3.55) follows

from the fact that Hermite polynomials are moments of slightly shifted Hermite poly-

nomials, as explained in [71]. In fact, the papers [71] and [72] contain more examples

of orthogonal polynomials which are moments, thus in particular implying Hankel de-

terminant evaluations whose entries are Laguerre polynomials, Meixner polynomials,

and Al-Salamā“Chihara polynomials. Hankel determinants where the entries are (clas-

sical) orthogonal polynomials are also considered in [77], where they are related to

Wronskians of orthogonal polynomials. In particular, there result Hankel determinant

evaluations with entries being Legendre, ultraspherical, and Laguerre polynomials [77,

(12.3), (14.3), (16.5), Ā§ 28], respectively. The reader is also referred to [103], where

illuminating proofs of these identities between Hankel determinants and Wronskians

are given, by using the fact that Hankel determinants can be seen as certain Schur

functions of rectangular shape, and by applying a ā˜master identityā™ of Turnbull [178,

p. 48] on minors of a matrix. (The evaluations (3.52), (3.55) and (3.56) can be found in

[103] as well, as corollaries to more general results.) Alternative proofs of (3.52), (3.54)

and (3.55) can be found in [141], see also [139] and [140].

Clearly, to prove (3.56)ā“(3.58) one would proceed in the same way as in Section 2.7.

(Identity (3.58) is in fact the evaluation (2.38) that we derived in Section 2.7.) The

evaluations (3.59) and (3.60) are equivalent to (3.58), because the matrix underlying

the determinant in (3.58) has a checkerboard pattern (recall that Bernoulli numbers

with odd indices are zero, except for B1 ), and therefore decomposes into the prod-

uct of a determinant of the form (3.59) and a determinant of the form (3.60). Very

interestingly, variations of (3.56)ā“(3.60) arise as normalization constants in statistical

mechanics models, see e.g. [14, (4.36)], [32, (4.19)], and [108, App. A.5]. A common

generalization of (3.56)ā“(3.58) can be found in [51, Sec. 5]. Strangely enough, it was

needed there in the enumeration of rhombus tilings.

48 C. KRATTENTHALER

In view of Section 2.7, any continued fraction expansion of the form (2.30) gives rise

to a Hankel determinant evaluation. Thus, many more Hankel determinant evaluations

follow e.g. from work by Rogers [151], Stieltjes [171, 172], Flajolet [44], Han, Randri-

anarivony and Zeng [65, 64, 142, 143, 144, 145, 146, 201], Ismail, Masson and Valent

[70, 73] or Milne [113, 114, 115, 116], in particular, evaluations of Hankel determinant

featuring Euler numbers with odd indices (these are given through the generating func-

tion tan z = ā E2k+1 z 2k+1/(2k + 1)!), Genocchi numbers, q- and other extensions of

k=0

Catalan, Euler and Genocchi numbers, and coeļ¬cients in the power series expansion

of Jacobi elliptic functions. Evaluations of the latter type played an important role in

Milneā™s recent beautiful results [113, 114] on the number of representations of integers

as sums of m-th powers (see also [108, App. A.5]).

For further evaluations of Hankel determinants, which apparently do not follow from

known results about continued fractions or orthogonal polynomials, see [68, Prop. 14]

and [51, Sec. 4].

Next we state two charming applications of Lemma 16 (see [189]).

Theorem 53. Let x be a nonnegative integer. For any nonnegative integer n there

hold

n+1

x (2)

(xi)!

det S(xi + j, xi) = (3.61)

(xi + j)! 2

0ā¤i,jā¤n

where S(m, k) is a Stirling number of the second kind (the number of partitions of an

m-element set into k blocks; cf. [166, p. 33]), and

n+1

x (2)

(xi)!

s(xi + j, xi) = ā’

det , (3.62)

(xi + j)! 2

0ā¤i,jā¤n

where s(m, k) is a Stirling number of the ļ¬rst kind (up to sign, the number of permu-

tations of m elements with exactly k cycles; cf. [166, p. 18]).

Theorem 54. Let Aij denote the number of representations of j as a sum of i squares

of nonnegative integers. Then det0ā¤i,jā¤n (Aij ) = 1 for any nonnegative integer n. The

same is true if āsquaresā is replaced by ācubes,ā etc.

After having seen so many determinants where rows and columns are indexed by

integers, it is time for a change. There are quite a few interesting determinants whose

rows and columns are indexed by (other) combinatorial objects. (Actually, we already

encountered one in Conjecture 49.)

We start by a determinant where rows and columns are indexed by permutations.

Its beautiful evaluation was obtained at roughly the same time by Varchenko [184] and

Zagier [193].

Theorem 55. For any positive integer n there holds

n

inv(ĻĻā’1 )

n

(1 ā’ q i(iā’1))( i )(iā’2)! (nā’i+1)! ,

det q = (3.63)

Ļ,ĻāSn

i=2

where Sn denotes the symmetric group on n elements.

ADVANCED DETERMINANT CALCULUS 49

This determinant evaluation appears in [193] in the study of certain models in inļ¬nite

statistics. However, as Varchenko et al. [20, 153, 184] show, this determinant evaluation

is in fact just a special instance in a whole series of determinant evaluations. The

latter papers give evaluations of determinants corresponding to certain bilinear forms

associated to hyperplane arrangements and matroids. Some of these bilinear forms

are relevant to the study of hypergeometric functions and the representation theory

of quantum groups (see also [185]). In particular, these results contain analogues of

(3.63) for all ļ¬nite Coxeter groups as special cases. For other developments related to

Theorem 55 (and diļ¬erent proofs) see [36, 37, 40, 67], tying the subject also to the

representation theory of the symmetric group, to noncommutative symmetric functions,

and to free Lie algebras, and [109]. For more remarkable determinant evaluations related

to hyperplane arrangements see [39, 182, 183]. For more determinant evaluations related

to hypergeometric functions and quantum groups and algebras, see [175, 176], where

determinants arising in the context of Knizhnik-Zamolodchikov equations are computed.

The results in [20, 153] may be considered as a generalization of the Shapovalov de-

terminant evaluation [159], associated to the Shapovalov form in Lie theory. The latter

has since been extended to Kacā“Moody algebras (although not yet in full generality),

see [31].

There is a result similar to Theorem 55 for another prominent permutation statistics,

MacMahonā™s major index. (The major index maj(Ļ) is deļ¬ned as the sum of all positions

of descents in the permutation Ļ, see e.g. [46].)

Theorem 56. For any positive integer n there holds

n

maj(ĻĻā’1 )

(1 ā’ q i )n! (iā’1)/i.

det q = (3.64)

Ļ,ĻāSn

i=2

As Jeanā“Yves Thibon explained to me, this determinant evaluation follows from

results about the descent algebra of the symmetric group given in [95], presented

there in an equivalent form, in terms of noncommutative symmetric functions. For

the details of Thibonā™s argument see Appendix C. Also the bivariate determinant

ā’1 ā’1

detĻ,ĻāSn xdes(ĻĻ ) q maj(ĻĻ ) seems to possess an interesting factorization.

The next set of determinant evaluations shows determinants where the rows and

columns are indexed by set partitions. In what follows, the set of all partitions of

{1, 2, . . . , n} is denoted by Ī n . The number of blocks of a partition Ļ is denoted by

bk(Ļ). A partition Ļ is called noncrossing, if there do not exist i < j < k < l such

that both i and k belong to one block, B1 say, while both j and l belong to another

block which is diļ¬erent from B1 . The set of all noncrossing partitions of {1, 2, . . . , n}

is denoted by NCn . (For more information about noncrossing partitions see [160].)

Further, poset-theoretic, notations which are needed in the following theorem are:

Given a poset P , the join of two elements x and y in P is denoted by x āØP y, while

the meet of x and y is denoted by x ā§P y. The characteristic polynomial of a poset

P is written as ĻP (q) (that is, if the maximum element of P has rank h and Āµ is the

Ė , where Ė stands for the

hā’rank(p)

MĀØbius function of P , then ĻP (q) :=

o pāP Āµ(0, p)q 0

minimal element of P ). The symbol ĻP (q) denotes the reciprocal polynomial q h ĻP (1/q)

Ė

ā—

of ĻP (q). Finally, P is the order-dual of P .

50 C. KRATTENTHALER

Theorem 57. Let n be a positive integer. Then

n

(n)B(nā’i)

i

bk(Ļā§Ī n Ī³)

det q = q ĻĪ ā— (q)

Ėi , (3.65)

Ļ,Ī³āĪ n

i=1

where B(k) denotes the k-th Bell number (the total number of partitions of a k-element

set; cf. [166, p. 33]). Furthermore,

n

S(n,i)

q bk(ĻāØĪ n Ī³) =

det q ĻĪ i (q) , (3.66)

Ļ,Ī³āĪ n

i=1

where S(m, k) is a Stirling number of the second kind (the number of partitions of an

m-element set into k blocks; cf. [166, p. 33]). Next,

n

(2nā’1ā’i)

2nā’1

= q( n )

nā’1

bk(Ļā§NCn Ī³)

det q ĻNCi (q)

Ė , (3.67)

Ļ,Ī³āNCn

i=1

and

n

(2nā’1ā’i)

2n

1

() nā’1

bk(ĻāØNCn Ī³)

det q =q ĻNCi (q) , (3.68)

n+1 n

Ļ,Ī³āNCn

i=1

Finally,

ā 2n

i+1

( )

nā’1

Ui+1 ( q/2) n nā’1ā’i

2nā’1

= q( ) ā

bk(ĻāØĪ n Ī³)

det q , (3.69)

n

qUiā’1 ( q/2)

Ļ,Ī³āNCn

i=1

j mā’j

(2x)mā’2j is the m-th Chebyshev polynomials of the

where Um (x) := jā„0 (ā’1) j

second kind.

The evaluations (3.65)ā“(3.68) are due to Jackson [75]. The last determinant eval-

uation, (3.69), is the hardest among those. It was proved independently by Dahab

[33] and Tutte [181]. All these determinants are related to the so-called Birkhoļ¬ā“Lewis

equation from chromatic graph theory (see [33, 180] for more information).

A determinant of somewhat similar type appeared in work by Lickorish [104] on 3-

manifold invariants. Let NCmatch(2n) denote the set of all noncrossing perfect match-

ings of 2n elements. Equivalently, NCmatch(2n) can be considered as the set of all

noncrossing partitions of 2n elements with all blocks containing exactly 2 elements.

Lickorish considered a bilinear form on the linear space spanned by NCmatch(2n). The

corresponding determinant was evaluated by Ko and Smolinsky [80] using an elegant

recursive approach, and independently by Di Francesco [47], whose calculations are

done within the framework of the Temperleyā“Lieb algebra (see also [49]).

Theorem 58. For Ī±, Ī² ā NCmatch(2n), let c(Ī±, Ī²) denote the number of connected

components when the matchings Ī± and Ī² are superimposed. Equivalently, c(Ī±, Ī²) =

bk(Ī± āØĪ 2n Ī²). For any positive integer n there holds

n

c(Ī±,Ī²)

Ui (q/2)a2n,2i ,

det q = (3.70)

Ī±,Ī²āNCmatch(2n)

i=1

where Um (q) is the Chebyshev polynomial of the second kind as given in Theorem 57,

and where a2n,2i = c2n,2i ā’ c2n,2i+2 with cn,h = (nā’h)/2 ā’ (nā’h)/2ā’1 .

n n

ADVANCED DETERMINANT CALCULUS 51

Di Francesco [47, Theorem 2] does also provide a generalization to partial matchings,

and in [48] a generalization in an SU(n) setting, the previously mentioned results being

situated in the SU(2) setting. While the derivations in [47] are mostly combinatorial,

the derivations in [48] are based on computations in quotients of type A Hecke algebras.

There is also an interesting determinant evaluation, which comes to my mind, where

rows and columns of the determinant are indexed by integer partitions. It is a result

due to Reinhart [147]. Interestingly, it arose in the analysis of algebraic diļ¬erential

equations.

In concluding, let me attract your attention to other determinant evaluations which

I like, but which would take too much space to state and introduce properly.

For example, there is a determinant evaluation, conjectured by Good, and proved by

Goulden and Jackson [60], which arose in the calculation of cumulants of a statistic anal-

ogous to Pearsonā™s chi-squared for a multinomial sample. Their method of derivation

is very combinatorial, in particular making use of generalized ballot sequences.

Determinants arising from certain raising operators of sl(2)-representations are pre-

sented in [136]. As special cases, there result beautiful determinant evaluations where

rows and columns are indexed by integer partitions and the entries are numbers of

standard Young tableaux of skew shapes.

In [84, p. 4] (see also [192]), an interesting mixture of linear algebra and combinatorial

matrix theory yields, as a by-product, the evaluation of the determinant of certain

incidence mappings. There, rows and columns of the relevant matrix are indexed by all

subsets of an n-element set of a ļ¬xed size.

As a by-product of the analysis of an interesting matrix in quantum information

theory [93, Theorem 6], the evaluation of a determinant of a matrix whose rows and

columns are indexed by all subsets of an n-element set is obtained.

Determinant evaluations of q-hypergeometric functions are used in [177] to compute

q-Selberg integrals.

And last, but not least, let me once more mention the remarkable determinant evalu-

ation, arising in connection with holonomic q-diļ¬erence equations, due to Aomoto and

Kato [11, Theorem 3], who thus proved a conjecture by Mimachi [118].

Appendix A: A word about guessing

The problem of guessing a formula for the generic element an of a sequence (an )nā„0

out of the ļ¬rst few elements was present at many places, in particular this is crucial for

a successful application of the āidentiļ¬cation of factorsā method (see Section 2.4) or of

LU-factorization (see Section 2.6). Therefore some elaboration on guessing is in order.

First of all, as I already claimed, guessing can be largely automatized. This is due to

the following tools11 :

1. Superseeker, the electronic version of the āEncyclopedia of Integer Sequencesā

[162, 161] by Neil Sloane and Simon Plouļ¬e (see Footnote 2 in the Introduction),

11

In addition, one has to mention Frank Garvanā™s qseries [54], which is designed for guessing and

computing within the territory of q-series, q-products, eta and theta functions, and the like. Procedures

like prodmake or qfactor, however, might also be helpful for the evaluation of āq-determinantsā. The

package is available from http://www.math.ufl.edu/˜frank/qmaple.html.

52 C. KRATTENTHALER

2. gfun by Bruno Salvy and Paul Zimmermann and Mgfun by Frederic Chyzak (see

Footnote 3 in the Introduction),

3. Rate by the author (see Footnote 4 in the Introduction).

If you send the ļ¬rst few elements of your sequence to Superseeker then, if it overlaps

with a sequence that is stored there, you will receive information about your sequence

such as where your sequence already appeared in the literature, a formula, generating

function, or a recurrence for your sequence.

The Maple package gfun provides tools for ļ¬nding a generating function and/or a

recurrence for your sequence. (In fact, Superseeker does also automatically invoke

features from gfun.) Mgfun does the same in a multidimensional setting.

Within the āhypergeometric paradigm,ā the most useful is the Mathematica pro-

gram Rate (āRate!ā is German for āGuess!ā), respectively its Maple equivalent GUESS.

Roughly speaking, it allows to automatically guess āclosed formsā.12 The program

is based on the observation that any āclosed formā sequence (an )nā„0 that appears

within the āhypergeometric paradigmā is either given by a rational expression, like

an = n/(n + 1), or the sequence of successive quotients (an+1 /an )nā„0 is given by a ratio-

nal expression, like in the case of central binomial coeļ¬cients an = 2n , or the sequence

n

of successive quotients of successive quotients ((an+2 /an+1 )/(an+1 /an ))nā„0 is given by

a rational expression, like in the case of the famous sequence of numbers of alternating

sign matrices (cf. the paragraphs following (3.9), and [18, 19, 111, 148, 97, 198, 199]

for information on alternating sign matrices),

nā’1

(3i + 1)!

an = , (A.1)

(n + i)!

i=0

etc. Given enough special values, a rational expression is easily found by rational

interpolation.

This is implemented in Rate. Given the ļ¬rst m terms of a sequence, it takes the

ļ¬rst m ā’ 1 terms and applies rational interpolation to these, then it applies rational

interpolation to the successive quotients of these m ā’ 1 terms, etc. For each of the

obtained results it is tested if it does also give the m-th term correctly. If it does, then

the corresponding result is added to the output, if it does not, then nothing is added

to the output.

Here is a short demonstration of the Mathematica program Rate. The output shows

guesses for the i0-th element of the sequence.

In[1]:= rate.m

In[2]:= Rate[1,2,3]

Out[2]= {i0}

In[3]:= Rate[2/3,3/4,4/5,5/6]

12

Commonly, by āclosed formā (āNICEā in Zeilbergerā™s āterminologyā) one means an expression

which is built by forming products and quotients of factorials. A strong indication that you encounter

a sequence (an )nā„0 for which a āclosed formā exists is that the prime factors in the prime factorization

of an do not grow rapidly as n becomes larger. (In fact, they should grow linearly.)

ADVANCED DETERMINANT CALCULUS 53

1 + i0

Out[3]= {------}

2 + i0

Now we try the central binomial coeļ¬cients:

In[4]:= Rate[1,2,6,20,70]

-1 + i0

2 (-1 + 2 i1)

Out[4]= { -------------}

i1

i1=1

It needs the ļ¬rst 8 values to guess the formula (A.1) for the numbers of alternating sign

matrices:

In[5]:= Rate[1,2,7,42,429,7436,218348,10850216]

-1 + i0 -1 + i1

3 (2 + 3 i2) (4 + 3 i2)

Out[5]= { 2( -----------------------)}

4 (1 + 2 i2) (3 + 2 i2)

i1=1 i2=1

However, what if we encounter a sequence where all these nice automatic tools fail?

Here are a few hints. First of all, it is not uncommon to encounter a sequence (an )nā„0

which has actually a split deļ¬nition. For example, it may be the case that the subse-

quence (a2n )nā„0 of even-numbered terms follows a āniceā formula, and that the subse-

quence (a2n+1)nā„0 of odd-numbered terms follows as well a ānice,ā but diļ¬erent, formula.

Then Rate will fail on any number of ļ¬rst terms of (an )nā„0 , while it will give you some-

thing for suļ¬ciently many ļ¬rst terms of (a2n)nā„0 , and it will give you something else

for suļ¬ciently many ļ¬rst terms of (a2n+1 )nā„0 .

Most of the subsequent hints apply to a situation where you encounter a sequence

p0 (x), p1(x), p2 (x), . . . of polynomials pn (x) in x for which you want to ļ¬nd (i.e., guess)

a formula. This is indeed the situation in which you are generally during the guessing

for āidentiļ¬cation of factors,ā and also usually when you perform a guessing where a

parameter is involved.

To make things concrete, let us suppose that the ļ¬rst 10 elements of your sequence

of polynomials are

54 C. KRATTENTHALER

1 1

6 + 31x ā’ 15x2 + 20x3 , 12 ā’ 58x + 217x2 ā’ 98x3 + 35x4 ,

1, 1 + 2x, 1 + x + 3x2 ,

6 12

1 1

20+508xā’925x2 +820x3 ā’245x4 +42x5 , 120ā’8042x+20581x2 ā’17380x3 +7645x4 ā’1518x5 +154x6 ,

20 120

1

1680 + 386012x ā’ 958048x2 + 943761x3 ā’ 455455x4 + 123123x5 ā’ 17017x6 + 1144x7 ,

1680

1

20160ā’15076944x+40499716x2 ā’42247940x3 +23174515x4 ā’7234136x5 +1335334x6 ā’134420x7 +6435x8 ,

20160

1

181440 + 462101904x ā’ 1283316876x2 + 1433031524x3 ā’ 853620201x4 + 303063726x5

181440

ā’ 66245634x6 + 8905416x7 ā’ 678249x8 + 24310x9 , . . . (A.2)

You may of course try to guess the coeļ¬cients of powers of x in these polynomials.

But within the āhypergeometric paradigmā this does usually not work. In particular,

that does not work with the above sequence of polynomials.

A ļ¬rst very useful idea is to guess through interpolation. (For example, this is what

helped to guess coeļ¬cients in [43].) What this means is that, for each pn (x) you try to

ļ¬nd enough values of x for which pn (x) appears to be āniceā (the prime factorization

of pn (x) has small prime factors, see Footnote 12). Then you guess these special eval-

uations of pn (x) (by, possibly, using Rate or GUESS), and, by interpolation, are able to

write down a guess for pn (x) itself.

Let us see how this works for our sequence (A.2). A few experiments will convince

you that pn (x), 0 ā¤ n ā¤ 9 (this is all we have), appears to be āniceā for x = 0, 1, . . . , n.

Furthermore, using Rate or GUESS, you will quickly ļ¬nd that, apparently, pn (e) = 2n+e e

for e = 0, 1, . . . , n. Therefore, as it also appears to be the case that pn (x) is of degree

n, our sequence of polynomials should be given (using Lagrange interpolation) by

n

2n + e x(x ā’ 1) Ā· Ā· Ā· (x ā’ e + 1)(x ā’ e ā’ 1) Ā· Ā· Ā· (x ā’ n)

pn (x) = . (A.3)

e(e ā’ 1) Ā· Ā· Ā· 1 Ā· (ā’1) Ā· Ā· Ā· (e ā’ n)

e

e=0

Another useful idea is to try to expand your polynomials with respect to a āsuitableā

basis. (For example, this is what helped to guess coeļ¬cients in [30] or [94, e.g., (3.15),

(3.33)].) Now, of course, you do not know beforehand what āsuitableā could be in your

situation. Within the āhypergeometric paradigmā candidates for a suitable basis are

always more or less sophisticated shifted factorials. So, let us suppose that we know

that we were working within the āhypergeometric paradigmā when we came across

the example (A.2). Then the simplest possible bases are (x)k , k = 0, 1, . . . , or (ā’x)k ,

k = 0, 1, . . . . It is just a matter of taste, which of these to try ļ¬rst. Let us try the

latter. Here are the expansions of p3 (x) and p4 (x) in terms of this basis (actually, in

terms of the equivalent basis x , k = 0, 1, . . . ):

k

6 + 31x ā’ 15x2 + 20x3 = 1 + 6 x x x

1

+ 15 + 20 ,

6 1 2 3

12 ā’ 58x + 217x2 ā’ 98x3 + 35x4 = 1 + 8 x x x x

1

+ 28 + 56 + 70 .

12 1 2 3 4

I do not know how you feel. For me this is enough to guess that, apparently,

n

2n x

pn (x) = .

k k

k=0

ADVANCED DETERMINANT CALCULUS 55

(Although this is not the same expression as in (A.3), it is identical by means of a

3 F2 -transformation due to Thomae, see [55, (3.1.1)]).

As was said before, we do not know beforehand what a āsuitableā basis is. Therefore

you are advised to get as much a priori information about your polynomials as possible.

For example, in [28] it was āknownā to the authors that the result which they wanted

to guess (before being able to think about a proof) is of the form (NICE PRODUCT) Ć—

(IRREDUCIBLE POLYNOMIAL). (I.e., experiments indicated that.) Moreover, they

knew that their (IRREDUCIBLE POLYNOMIAL), a polynomial in m, pn (m) say,

would have the property pn (ā’m ā’ 2n + 1) = pn (m). Now, if we are asking ourselves

what a āsuitableā basis could be that has this property as well, and which is built

in the way of shifted factorials, then the most obvious candidate is (m + n ā’ k)2k =

(m + n ā’ k)(m + n ā’ k + 1) Ā· Ā· Ā· (m + n + k ā’ 1), k = 0, 1, . . . . Indeed, it was very easy

to guess the expansion coeļ¬cients with respect to this basis. (See Theorems 1 and 2

in [28]. The polynomials that I was talking about are represented by the expression in

big parentheses in [28, (1.2)].)

If the above ideas do not help, then I have nothing else to oļ¬er than to try some,

more or less arbitrary, manipulations. To illustrate what I could possibly mean, let us

again consider an example. In the course of working on [90], I had to guess the result

of a determinant evaluation (which became Theorem 8 in [90]; it is reproduced here

as Theorem 43). Again, the diļ¬cult part of guessing was to guess the āuglyā part of

the result. As the dimension of the determinant varied, this gave a certain sequence

pn (x, y) of polynomials in two variables, x and y, of which I display p4 (x, y):

In[1]:= VPol[4]

2 3 4 2 3 2

Out[1]= 6 x + 11 x +6x + x + 6 y - 10 x y - 6 x y - 4 x y + 11 y -

2 2 2 3 3 4

> 6xy +6x y +6y -4xy +y

(What I subsequently describe is the actual way in which the expression for pn (x, y) in

terms of the sum on the right-hand side of (3.38) was found.) What caught my eyes was

the part of the polynomial independent of y, x4 + 6x3 + 11x2 + 6x, which I recognized

as (x)4 = x(x + 1)(x + 2)(x + 3). For the fun of it, I subtracted that, just to see what

would happen:

In[2]:= Factor[%-x(x+1)(x+2)(x+3)]

2 3 2 2 2

Out[2]= y (6 - 10 x - 6 x -4x + 11 y - 6 x y + 6 x y+6y -4xy +

3

y)

Of course, a y factors. Okay, let us cancel that:

In[3]:= %/y

2 3 2 2 2 3

Out[3]= 6 - 10 x - 6 x -4x + 11 y - 6 x y + 6 x y+6y -4xy +y

56 C. KRATTENTHALER

One day I had the idea to continue in a āsystematicā manner: Let us subtract/add an

appropriate multiple of (x)3 ! Perhaps, āappropriateā in this context is to add 4(x)3 ,

because that does at least cancel the third powers of x:

In[4]:= Factor[%+4x(x+1)(x+2)]

2 2

Out[4]= (1 + y) (6 - 2 x + 6 x + 5 y - 4 x y + y )

I assume that I do not have to comment the rest:

In[5]:= %/(y+1)

2

Out[5]= 6 - 2 x + 6 x +5y-4xy+y

In[6]:= Factor[%-6x(x+1)]

Out[6]= (2 + y) (3 - 4 x + y)

In[7]:= %/(y+2)

Out[7]= 3 - 4 x + y

In[8]:= Factor[%+4x]

Out[8]= 3 + y

What this shows is that

p4 (x, y) = (x)4 ā’ 4(x)3 (y)1 + 6(x)2 (y)2 ā’ 4(x)1 (y)3 + (y)4 .

No doubt that, at this point, you would have immediately guessed (as I did) that, in

general, we āmustā have (compare (3.38))

n

n

(ā’1)k

pn (x, y) = (x)k (y)nā’k .

k

k=0

Appendix B: Turnbullā™s polarization of Bazinā™s theorem implies most of the

identities in Section 2.2

In this appendix we show that all the determinant lemmas from Section 2.2, with the

exception of Lemmas 8 and 9, follow from the evaluation of a certain determinant of

minors of a given matrix, an observation which I owe to Alain Lascoux. This evaluation,

due to Turnbull [179, p. 505], is a polarized version of a theorem of Bazin [119, II,

pp. 206ā“208] (see also [102, Sec. 3.1 and 3.4]).

For the statement of Turnbullā™s theorem we have to ļ¬x an n-rowed matrix A, in which

we label the columns, slightly unconventionally, by a2, . . . , am , b21, b31, b32, b41, . . . , bn,nā’1 ,

x1, x2, . . . , xn , for some m ā„ n, i.e., A is an n Ć— (n + m ā’ 1 + n ) matrix. Finally, let

2

[a, b, c, . . . ] denote the minor formed by concatenating columns a, b, c, . . . of A, in that

order.

ADVANCED DETERMINANT CALCULUS 57

Proposition 59. (Cf. [179, p. 505], [102, Sec. 3.4]). With the notation as explained

above, there holds

det [bj,1, bj,2 , . . . , bj,jā’1 , xi , aj+1 , . . . , am ]

1ā¤i,jā¤n

n

= [x1, x2, . . . , xn , an+1 , . . . , am ] [bj,1, bj,2, . . . , bj,jā’1 , aj , . . . , am ]. (B.1)

j=2

Now, in order to derive Lemma 3 from (B.1), we choose m = n and for A the matrix

a2 ... an b21 b31 b32 ... bn,nā’1 x1 x2 ... xn

ļ£« ļ£¶

1 ... 1 1 1 1 ... 1 1 1 ... 1

ļ£¬ ā’A2 X2 . . . X n ļ£·

ā’An ā’B2 ā’B2 ā’B3 ā’Bn

... ... X1

ļ£¬ ļ£·

ļ£¬ (ā’A2)2 . . . (ā’An )2 X2 . . . X n ļ£· ,

(ā’B2 )2 (ā’B2 )2 (ā’B3 )2 . . . (ā’Bn )2 2 2 2

X1

ļ£¬ ļ£·

ļ£. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ļ£ø

(ā’A2 )nā’1 . . . (ā’An )nā’1 (ā’B2 )nā’1 (ā’B2 )nā’1 (ā’B3 )nā’1 . . . (ā’Bn )nā’1 X1 nā’1 nā’1 nā’1

X2 . . . Xn

with the unconventional labelling of the columns indicated on top. I.e., column bst is

ļ¬lled with powers of ā’Bt+1 , 1 ā¤ t < s ā¤ n. With this choice of A, all the minors in (B.1)

are Vandermonde determinants. In particular, due to the Vandermonde determinant

evaluation (2.1), we then have for the (i, j)-entry of the determinant in (B.1)

[bj,1, bj,2, . . . , bj,jā’1 , xi, aj+1 , . . . , am]

j n

(Bs ā’ Bt ) (As ā’ At) (At ā’ Bs )

=

2ā¤s<tā¤j j+1ā¤s<tā¤n s=2 t=j+1

j

n

Ć— (Xi + As ) (Xi + Bs ),

s=j+1 s=2

which is, up to factors that only depend on the column index j, exactly the (i, j)-entry

of the determinant in (2.8). Thus, Turnbullā™s identity (B.1) gives the evaluation (2.8)

immediately, after some obvious simpliļ¬cation.

In order to derive Lemma 5 from (B.1), we choose m = n and for A the matrix

a2 ... an b21 b31

ļ£«

1 ... 1 1 1

ļ£¬ ā’A2 ā’ C/A2 ā’An ā’ C/An ā’B2,1 ā’ C/B2,1 ā’B3,1 ā’ C/B3,1

...

ļ£¬

ļ£¬ (ā’A2 ā’ C/A2 )2 . . . (ā’An ā’ C/An )2 (ā’B2,1 ā’ C/B2,1 ) (ā’B3,1 ā’ C/B3,1 )2

2

ļ£¬

ļ£. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ā’A2 ā’ C/A2 )nā’1 . . . (ā’An ā’ C/An )nā’1 (ā’B2,1 ā’ C/B2,1 )nā’1 (ā’B3,1 ā’ C/B3,1 )nā’1

b32 ... bn,nā’1 x1 ... xn

ļ£¶

1 ... 1 1 ... 1

Xn + C/Xn ļ£·

ā’B3,2 ā’ C/B3,2 ā’Bn,nā’1 ā’ C/Bn,nā’1

... X1 + C/X1 ... ļ£·

. . . (Xn + C/Xn )2 ļ£· .

(ā’B3,2 ā’ C/B3,2 ) . . . (ā’Bn,nā’1 ā’ C/Bn,nā’1 )

2 2 2

(X1 + C/X1 ) ļ£·

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ļ£ø

(ā’B3,2 ā’ C/B3,2 )nā’1 . . . (ā’Bn,nā’1 ā’ C/Bn,nā’1 )nā’1 (X1 + C/X1 )nā’1 . . . (Xn + C/Xn )nā’1

58 C. KRATTENTHALER

(In this display, the ļ¬rst line contains columns a2, . . . , b31 of A, while the second line

contains the remaining columns.) Again, with this choice of A, all the minors in (B.1)

are Vandermonde determinants. Therefore, by noting that (S + C/S) ā’ (T + C/T ) =

(S ā’ T )(C/S ā’ T )/(ā’T ), and by writing pjā’1 (X) for

jā’1

(X + Bj,s )(C/X + Bj,s ), (B.2)

s=1

we have for the (i, j)-entry of the determinant in (B.1)

(Bj,s + C/Bj,s ā’ Bj,t ā’ C/Bj,t )

[bj,1, bj,2, . . . , bj,jā’1 , xi, aj+1 , . . . , am] =

1ā¤s<tā¤jā’1

jā’1 n

Ć— (As + C/As ā’ At ā’ C/At ) (At + C/At ā’ Bj,s ā’ C/Bj,s )

j+1ā¤s<tā¤n s=1 t=j+1

jā’1

n

As ) Aā’1 ā’1

Ć— (Xi + As )(C/Xi + pjā’1 (Xi ) Bj,s

s

s=1

s=j+1

for the (i, j)-entry of the determinant in (B.1). This is, up to factors which depend

only on the column index j, exactly the (i, j)-entry of the determinant in (2.11). The

polynomials pjā’1 (X), j = 1, 2, . . . , n, can indeed be regarded as arbitrary Laurent poly-

nomials satisfying the conditions of Lemma 5, because any Laurent polynomial qjā’1 (X)

over the complex numbers of degree at most j ā’ 1 and with qjā’1 (X) = qjā’1 (C/X) can

be written in the form (B.2). Thus, Turnbullā™s identity (B.1) implies the evaluation

(2.11) as well.

Similar choices for A are possible in order to derive Lemmas 4, 6 and 7 (which are in

fact just limiting cases of Lemma 5) from Proposition 59.

Appendix C: Jean-Yves Thibonā™s proof of Theorem 56

Obviously, the determinant in (3.64) is the determinant of the linear operator

Kn (q) := ĻāSn q maj Ļ Ļ acting on the group algebra C[Sn ] of the symmetric group.

Thus, if we are able to determine all the eigenvalues of this operator, together with

their multiplicities, we will be done. The determinant is then just the product of all

the eigenvalues (with multiplicities).

The operator Kn (q) is also an element of Solomonā™s descent algebra (because permu-

tations with the same descent set must necessarily have the same major index). The

descent algebra is canonically isomorphic to the algebra of noncommutative symmetric

functions (see [56, Sec. 5]). It is shown in [95, Prop. 6.3] that, as a noncommutative

symmetric function, Kn (q) is equal to (q; q)n Sn (A/(1 ā’ q)), where Sn (B) denotes the

complete (noncommutative) symmetric function of degree n of some alphabet B.

The inverse element of Sn (A/(1 ā’ q)) happens to be Sn((1 ā’ q)A), i.e., Sn ((1 ā’ q)A) ā—

Sn(A/(1ā’q)) = Sn(A),13 with ā— denoting the internal multiplication of noncommutative

symmetric functions (corresponding to the multiplication in the descent algebra). This

n

is seen as follows. As in [95, Sec. 2.1] let us write Ļ(B; t) = nā„0 Sn (B)t for the

13

By deļ¬nition of the isomorphism between noncommutative symmetric functions and elements in

the descent algebra, Sn (A) corresponds to the identity element in the descent algebra of Sn .

ADVANCED DETERMINANT CALCULUS 59

generating function for complete symmetric functions of some alphabet B, and Ī»(B; t) =

n

nā„0 Īn (B)t for the generating function for elementary symmetric functions, which

are related by Ī»(B; ā’t)Ļ(B; t) = 1. Then, by [95, Def. 4.7 and Prop. 4.15], we have

Ļ((1 ā’ q)B; 1) = Ī»(B; ā’q)Ļ(B; 1). Let X be the ordered alphabet Ā· Ā· Ā· < q 2 < q < 1, so

that XA = A/(1 ā’ q). According to [95, Theorem 4.17], it then follows that

Ļ((1 ā’ q)A; 1) ā— Ļ(XA; 1) = Ļ((1 ā’ q)XA; 1) = Ī»(XA; ā’q)Ļ(XA; 1)

= Ī»(XA; ā’q)Ļ(XA; q)Ļ(A; 1) = Ļ(A; 1),

since by deļ¬nition of X, Ļ(XA; 1) is equal to Ļ(XA; q)Ļ(A; 1) (see [95, Def. 6.1]).

Therefore, Sn ((1 ā’ q)A) ā— Sn (XA) = Sn(A), as required.

Hence, we infer that Kn (q) is the inverse of Sn ((1 ā’ q)A)/(q; q)n.

The eigenvalues of Sn ((1 ā’ q)A) are given in [95, Lemma 5.13]. Their multiplicities

follow from a combination of Theorem 5.14 and Theorem 3.24 in [95], since the con-

struction in Sec. 3.4 of [95] yields idempotents eĀµ such that the commutative immage

of Ī±(eĀµ ) is equal to pĀµ /zĀµ . Explicitly, the eigenvalues of Sn ((1 ā’ q)A) are iā„1 (1 ā’ q Āµi ),

where Āµ = (Āµ1 , Āµ2 , . . . ) varies through all partitions of n, with corresponding multiplic-

ities n!/zĀµ , the number of permutations of cycle type Āµ, i.e., zĀµ = 1m1 m1! 2m2 m2! Ā· Ā· Ā· ,

where mi is the number of occurences of i in the partition Āµ, i = 1, 2, . . . . Hence, the

eigenvalues of Kn (q) are (q; q)n/ iā„1 (1 ā’ q Āµi ), with the same multiplicities.

Knowing all the eigenvalues of Kn (q) and their multiplicities explicitly, it is now not

extremely diļ¬cult to form the product of all these and, after a short calculation, recover

the right-hand side of (3.64).

Acknowledgements

I wish to thank an anonymous referee, Joris Van der Jeugt, Bernard Leclerc, Madan

Lal Mehta, Alf van der Poorten, Volker Strehl, Jean-Yves Thibon, Alexander Varchenko,

and especially Alain Lascoux, for the many useful comments and discussions which

helped to improve the contents of this paper considerably.

60 C. KRATTENTHALER

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ĀØ ĀØ

Institut fur Mathematik der Universitat Wien, Strudlhofgasse 4, A-1090 Wien, Aus-

tria.

e-mail: KRATT@Pap.Univie.Ac.At, WWW: http://radon.mat.univie.ac.at/People/kratt

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