{'fraction_non_alphanumeric': 0.06535709069665548, 'fraction_numerical': 0.02398860814955455, 'mean_word_length': 3.7591659426585577, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 20, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Thom polynomials and Schur functions I

We give the Thom polynomials for the singularities I_2,2 and A_3 associated with maps (C^n,0) ->(C^{n+k},0) with parameter k>=0. We give the Schur function expansions of these Thom polynomials. Moreover, for the singularities A_i (with any parameter k>= 0) we analyze the ``first approximation'' F^(i) to the Thom polynomial. Our computations combine the characterization of Thom polynomials via the ``method of restriction equations'' of Rimanyi et al. with the techniques of (super) Schur functions.


Introduction
The global behavior of singularities is governed by their Thom polynomials (cf. [27], [11], [1], [10], [24]). Knowing the Thom polynomial of a singularity η, denoted T η , one can compute the cohomology class represented by the η-points of a map. We do not attempt here to survey all activities related to computations of Thom polynomials -note that in Kleiman's survey [11], the reader can find a summary of an early work of Thom, Porteous, Ronga, Menn, Sergeraert, Lascoux, and Roberts on Thom polynomials presented in an algebro-geometric framework.
The way of obtaining the thought Thom polynomial is through the solution of a system of linear equations, which is fine when we want to find one concrete Thom polynomial, say, for a fixed k. However, if we want to find the Thom polynomials for a series of singularities, associated with maps (C • , 0) → (C •+k , 0) with k as a parameter, we have to solve simultaneously a countable family of systems of linear equations. As stated by Rimanyi in [24], p. 512 : "However, another challenge is to find Thom polynomials containing k as a parameter." We do it here for the restriction equations for the above mentioned singularities. In fact, the obtained functional equations in symmetric functions are of independent interest. The main novelty of the present paper over the previous articles on Thom polynomials, is an extensive use of Schur functions. Namely, instead of using Chern monomial expansions (as the authors of all previous papers constantly do), we use Schur function expansions. This puts a more transparent structure on computations of Thom polynomials. We hope that an expression for T I 2,2 (Theorem 16), as well as our expression for T A 3 (Theorem 22), provide a support of this claim. For example, we get in this way some recursive formulas (cf., e.g., Lemma 12) that are not so easy to find using other bases, in particular the Chern monomial basis. In fact, recursions play a prominent role in the formulas of the present paper, cf. Eqs. (53) and (75).
Another feature of using the Schur function expansions for Thom polynomials is that in all known to us cases (not only those treated in the present paper), all the coefficients are nonnegative. 1 To be more precise, we use here (the specializations of) supersymmetric Schur functions, also called super-S-functions or Schur functions in difference of alphabets together with their three basic properties: vanishing, cancella-tion and factorization, (cf. [3], [15], [20], [21], [16], [8], and [13]). These functions contain resultants among themselves. Their geometric significance was illuminated in the 80's in the author's study of polynomials supported on degeneracy loci (cf. [19]). In fact, in the present paper we use the point of view of that article to some extent. More precisely, given a morphism F → E of vector bundles, where rank(E) = m and rank(F ) = n, by a j-polynomial we understand a Z-linear combination of the Schur functions S I (E − F ), where partitions I are such that (n−j) m−j ⊂ I but (n−j+1) m−j+1 ⊂ I 2 .
In some sense, a j-polynomial is a "typical" polynomial supported on the jth degeneracy locus D j of the morphism (in the sense of [19], see also [8]).
A Thom polynomial is a sum of of such j-polynomials associated with the corresponding morphism of tangent bundles (cf. the next section). The polynomial T I 2,2 is a single j-polynomial whereas T A 3 for k > 0 is the sum of two j-polynomials (for two consecutive j's). We first determine its part related to the smaller rectangle, and then add necessary "corrections" related to the larger rectangle. For the singularities A i (any i), we describe the j-polynomial part of the Thom polynomial for the largest possible j, a sort of the "first approximation" to T A i (here, the corresponding rectangle is a single row of length k + 1).
We give here a complete proof of our formula for the Thom polynomial for the singularity I 2,2 . As for the singularities A i , we reprove the formulas of Thom and Ronga for A 1 and A 2 , and announce the Schur function expression for A 3 . We outline a proof in the Appendix. Another expression for A 3 in terms of monomials in the Chern classes was announced by Berczi, Feher, and Rimanyi in [2]. We also illustrate the methods used by reproving the result of Gaffney giving the Thom polynomial for A 4 with k = 0 (this was also done by Rimanyi [23] -our approach uses more extensively Schur functions). For any singularity A i , we describe the j-polynomial part (denoted by F (i) r ) of the Thom polynomial for the largest possible j. In our calculations we use extensively the λ-ring approach to symmetric functions developed mainly in Lascoux's book [13].

Recollections on Thom polynomials
Our main reference for this section is [24]. We start with recalling what we shall mean by a "singularity". Let k ≥ 0 be a fixed integer. By singularity we shall mean an equivalence class of stable germs (C • , 0) → (C •+k , 0), where • ∈ N, under the equivalence generated by right-left equivalence (i.e. analytic reparametrizations of the source and target) and suspension (by suspension of a germ κ we mean its trivial unfolding: (x, v) → (κ(x), v)).
We recall that the Thom polynomial T η of a singularity η is a polynomial in the formal variables c 1 , c 2 , . . . that after the substitution for a general map f : X → Y between complex analytic manifolds, evaluates the Poincaré dual [η(f )] of the cycle carried by the closure of By codimension of a singularity η, codim(η), we shall mean codim X (η(f )) for such an f . The concept of the polynomial T η comes from Thom's fundamental paper [27]. For a detailed discussion of the existence of Thom polynomials, see, e.g., [1]. Thom polynomials associated with group actions were studied in [10]. In fact, the above is the "usual case" with singularities in the region where moduli (continuous families) of singularities do not occur. This will be the case of the singularities studied in the present paper. Indeed, the codimension of all these singularities does not exceed 6k + 8, the lowest codimension when moduli of singularities start. According to Mather's classification, singularities are in one-to-one correspondence with finite dimensional C-algebras. We shall use the following notation: -I a,b (of Thom-Boardman type Σ 2 ) for stable germs with local algebra C[[x, y]]/(xy, x a + y b ), b ≥ a ≥ 2; -III a,b (of Thom-Boardman type Σ 2 ) for stable germs with local algebra C[[x, y]]/(xy, x a , y b ), b ≥ a ≥ 2 (here k ≥ 1).
Our computations of Thom polynomials for some of the above singularities, shall use the method which stems from a sequence of papers by Rimanyi et al. [25], [23], [24], [6], [2]. We sketch briefly this method, refering the interested reader for more details to these papers, the main references being the last three mentioned items.
Let k ≥ 0 be a fixed integer, and let η : (C • , 0) → (C •+k , 0) be a stable singularity with a prototype κ : (C n , 0) → (C n+k , 0). The maximal compact subgroup of the right-left symmetry group of κ will be denoted by G η . Even if Aut κ is much too large to be a finite dimensional Lie group, the concept of its maximal compact subgroup (up to conjugacy) can be defined in a sensible way (cf. [24]). It is clear that G η can be chosen so that images of its projections to the factors Diff(C n , 0) and Diff(C n+k , 0) are linear. Its representations via the projections on the source C n and the target C n+k will be denoted by λ 1 (η) and λ 2 (η). The vector bundles associated with the universal principal G η -bundle EG η → BG η using the representations λ 1 (η) and λ 2 (η) will be called E ′ η and E η . The total Chern class of the singularity η is defined in H • (BG η ; Z) by .
The Euler class of η is defined in In the following theorem we collect the information from [24], Theorem 2.4 and [6], Theorem 3.5, needed for the calculations in the present paper.
To use this method of determining the Thom polynomials for singularities, one needs their classification, see, e.g., [4].
In the forthcoming sections, we shall use these equations to compute Thom polynomials. Sometimes it will be convenient not to work with the whole maximal compact subgroup G η but with its suitable subgroup; this subgroup should be, however, as "close" to G η as possible (cf. [24], p. 502).
Being challenged by [24], p. 512 and especially [2], we shall find Thom polynomials containing k as a parameter -this seems to be a (much) more difficult task than computing Thom polynomials for separate values of k, because one must solve simultaneously a countable family of systems of linear equations.
To effectively use Theorem 1 we need to study the maximal compact subgroups of singularities. We recall the following recipe from [24] pp. 505-507. Let η be a singularity whose prototype is κ : (C n , 0) → (C n+k , 0). The germ κ is the miniversal unfolding of another germ β : (C m , 0) → (C m+k , 0) with dβ = 0. The group G η is a subgroup of the maximal compact subgroup of the algebraic automorphism group of the local algebra Q η of η times the unitary group U (k−d), where d is the difference between the minimal number of relations and the number of generators of Q η . With β well chosen, G η acts as right-left symmetry group on β with representations µ 1 and µ 2 . The representations λ 1 and λ 2 are where µ V is the representation of G η on the unfolding space V = C n−m given, for α ∈ V and (ϕ, ψ) ∈ G η , by For example, for the singularity of type where ρ j denotes the standard representation of the unitary group U (j).
Hence we obtain assertion (i) of the following Proposition 2 (i) Let η = A i ; for any k, writing x and y 1 ,. . . , y k for the Chern roots of the universal bundles on BU (1) and BU (k), , and writing x 1 , x 2 for the Chern roots of the universal bundles on two copies of BU (1) and on BU (k), , and writing x 1 , x 2 and y 1 , . . . , y k−1 for the Chern roots of the universal bundles on BU (2) and BU (k−1), These assertions are obtained, in a standard way, following the instructions of [24], Sect. 4.

Notational conventions Rather than the Chern classes
for the alphabets of the Chern roots A, B of T X * and T Y * . The reader will find in the next section a summary of algebraic properties of the functions S i (A − B), or, more generally, Schur functions S I (A − B) (I runs over sequences of integers) widely used in the present paper. Moreover, it will be more handy to use, instead of k, a "shifted" parameter r := k + 1 .
Sometimes, we shall write the Thom polynomial as T η r to emphasize its dependence on r. So, e.g., in our notation, the Thom polynomial for the singularity A 1 with codim(A 1 ) = r for r ≥ 1 (in general, codim(A i ) = ri), will be : In general, a Thom polynomial in terms of the c i 's (in those papers) will be written here as a linear combination of Schur functions obtained by changing each c i to S i and expanding in the Schur function basis. Another example is, for r = 1, the Thom polynomial for A 2 : c 2 1 + c 2 rewritten in the present notation as

Recollections on Schur functions
In this section we collect needed notions related to symmetric functions and prove a useful Lemma 10. We adopt the point of view of [13] for what concerns symmetric functions. Namely, given a commutative ring, we treat symmetric functions as operators acting on the ring. (Here, these commutative rings are mostly Z-algebras generated by the Chern roots of the vector bundles from Proposition 2.) Definition 3 By an alphabet A, we understand a (finite) multi-set of elements in a commutative ring.

Definition 4 Given two alphabets
The notation A − B is compatible with the multiplication of series: the sum A + A ′ denoting the union of two alphabets A and A ′ .

Convention 5
We shall often identify an alphabet A = {a 1 , . . . , a m } with the sum a 1 + · · · + a m and perform usual algebraic operations on such elements. For example, Ab will denote the alphabet (a 1 b, . . . , a m b). We will give priority to the algebraic notation over the set-theoretic one. In fact, in the following, we shall use mostly alphabets of variables.
We have (A+C)−(B+C) = A−B, and this corresponds to simplification of the common factor for the rational series: Definition 6 Given a sequence I = (i 1 , i 2 , . . . , i k ) ∈ Z k , and alphabets A and B, the Schur function S I (A−B) is We shall mostly use the case when I is a partition I = (0 ≤ i 1 ≤ · · · ≤ i k ). In fact, by permuting the columns we see that any determinant of the form (19) is either zero or is, up to sign, such a determinant indexed by a partition. These functions are often called supersymmetric Schur functions or Schur functions in difference of alphabets. Their properties were studied, among others, in [3], [15], [20], [21], [16], [8], and [13]; in the present paper, we shall use the notation and conventions from this last item). For example, By Eq. (18), we get the following cancellation property: In the following, we shall identify partitions with their Young diagrams, as is customary.
We record the following property (loc.cit.), justifying the notational remark from the end of Section 2; for a partition I, where J is the conjugate partition of I (i.e. the consecutive rows of J are equal to the corresponding columns of I), and A * denotes the alphabet Fix two positive integers m and n. We shall say that a partition I = Pictorially, this means that the Young diagram of I is contained in the "tickened" hook: We record the following vanishing property. Given alphabets A and B of cardinalities m and n, if a partition I is not contained in the (m, n)-hook, then (loc.cit.): For example, because 4569 is not contained in the (2, 4)-hook.
In fact, we have the following result (loc.cit.). (They form a Z-basis of the Abelian group of the so-called "supersymmetric functions" (loc.cit.).) In the present paper by a symmetric function, we shall mean a Z-linear combination of the operators S I (•). By the degree of such a symmetric function, we shall mean the largest weight |I| of a partition I involved in its Schur function expansion.

Theorem 7 If
The following useful convention stems from Lascoux's paper [14].

Convention 8
We may need to specialize a letter to 2, but this must not be confused with taking two copies of 1. To allow one, nevertheless, specializing a letter to an (integer, or even complex) number r inside a symmetric function, without introducing intermediate variables, we write r for this specialization. Boxes have to be treated as single variables. For example, S i (2) = i+1 2 but S i ( 2 ) = 2 i . A similar remark applies to Z-linear combinations of variables. We have S 2 (X 2 ) = x 2 1 +x 1 x 2 +x 2 2 but S 2 ( x 1 +x 2 ) = x 2 1 +2x 1 x 2 +x 2 2 , S 11 (X 2 ) = x 1 x 2 but S 11 ( x 1 +x 2 ) = 0, S 2 (3x) = 6x 2 but S 2 ( 3x ) = 9x 2 etc.
This terminology is justified by the fact that R(A, B) is the classical resultant of the polynomials R(x, A) and R(x, B). For example, Eq. (10) can be rewritten as and Eq. (14) is We have (loc.cit.) where the sum is over all partitions I ⊂ (n m ). When I is contained in the (m, n)-hook and at the same time I contains the rectangle (n m ), then we have the following factorization property (loc.cit.): for partitions I = (i 1 , . . . , i m ) and J = (j 1 , . . . , j k ), We now pass to the following function F . Fix positive integers m and n. For an alphabet A of cardinality m, we set where the sum is over partitions I = (i 1 ≤ i 2 ≤ · · · ≤ i m ≤ n).
Lemma 10 For a variable x and an alphabet B of cardinality n, Proof. For a fixed partition I = (i 1 ≤ i 2 ≤ · · · ≤ i m ≤ n), it follows from the factorization property (25)  The lemma has been proved. ✷ We end this section, with an example illustrating the way we shall use symmetric functions in our computations. We consider the singularity III 2,2 (with parameter r ≥ 2) whose codimension is 2r + 2. We know that T III 2,2 is equal to the Thom polynomial for Σ 2 , and the latter polynomial was computed in [18].
But let us apply directly Theorem 1 to the singularity III 2,2 . By virtue of Proposition 2, for r ≥ 3, the equations characterizing the Thom polynomial for III 2,2 are: and additionally, Here, without loss of generality, we assume that x, x 1 , x 2 , and B r−1 are variables, and P (•) denotes a symmetric function. Indeed, the singularities = III 2,2 of codimension ≤ codim(III 2,2 ) are: A 0 , A 1 , A 2 . For r = 2 we must add the vanishing imposed by A 3 which (similarly to III 2,2 ) is of codimension 6 (this is the only exception): Since the partition (r+1, r+1) is not contained in the (1, r)-hook, for P = S r+1,r+1 (•) we get the vanishings (28) and (30). Moreover, Eq. (29) is satisfied for this P because These equations characterize the Thom polynomial for III 2,2 , and hence this polynomial is equal to S r+1,r+1 in agreement with [18]. In the forthcoming computations, however, the method of restriction equations from Theorem 1 will play a principal role. 4 Thom polynomial for I 2,2 The codimension of I 2,2 (for parameter r ≥ 1) is 3r + 1. For r = 1, the Thom polynomial for I 2,2 is S 22 (cf. [18]). From now on, we shall assume that r ≥ 2. For r = 2, the Thom polynomial for I 2,2 is (cf. [24]): S 133 + 3S 34 .
By virtue of Proposition 2, the equations from Theorem 1 characterizing the Thom polynomial for I 2,2 are: and Here, without loss of generality, we assume that x, x 1 , x 2 , and B r−1 are variables. Moreover, P (•) denotes a symmetric function. For the remainder of this paper, we set Then, additionally, for variables x 1 , x 2 and an alphabet B r−2 , we have the vanishing imposed by III 2,2 : Indeed, the singularities = I 2,2 with codimension ≤ codim(I 2,2 ) are: A 0 , A 1 , A 2 , III 2,2 . For r ≥ 1, we set P r (•) := T Lemma 11 (i) A partition appearing nontrivially in the Schur function expansion of P r contains the rectangular partition (r + 1, r + 1).
(ii) A partition appearing nontrivially in the Schur function expansion of P r has at most three parts.
Proof. (i) This follows from the fact that the singularity I 2,2 belongs to the Thom-Boardman singularity Σ 2 .
(ii) We can assume that r ≥ 3. In addition to information contained in (i), we shall use Eq. (34): By virtue of (i), we can use factorization property (25) to all summands of (we assume that α I = 0). We divide each summand of this last polynomial by the resultant R(X 2 , D + B r−2 ) .
Suppose that the resulting factor of S I is: cf. (25). Since |I| = 3r + 1, we have Now, let us assume that I has more than 3 parts, that is J has more than 2 parts. This assumption (together with the inequality (38)) implies that S J (−B r−2 ) = 0 (B r−2 is an alphabet of variables). Expanding (37), we get among summands the following one of largest possible degree |J| in B r−2 : Take in the sum the (sub)sum of all the nonzero summands of the form (37) with the largest possible weight of J. Since Schur polynomials are independent this subsum is nonzero and moreover it is Z-linearly independent of other summands both in the sum indexed by partitions with ≥ 3 parts, and as well as in that indexed by partitions with 2 parts (this last sum does not depend on B r−2 ). Hence, there is no Z-linear combination of S I 's which involve nontrivially I with more than three parts and possibly also those with 3 and 2 parts, that satisfies Eq. (34). Assertion (ii) has been proved. ✷ The following lemma gives a recursive description of P r . Denote by τ the linear endomorphism on the Z-module of Schur functions corresponding to partitions of length ≤ 3 that sends a Schur function S i 1 ,i 2 ,i 3 to S i 1 +1,i 2 +1,i 3 +1 . Let P o r denote the sum of those terms in the Schur function expansion of P r which correspond to partitions of length ≤ 2. Note that P o 1 = S 22 . Lemma 12 With this notation, for r ≥ 2, we have the following recursive equation: Proof. Write where J have 2 parts and K = (k 1 , k 2 , k 3 ) have 3 parts (we assume that α I = 0). We set and our goal is to show that Q = P r−1 . Since a partition I appearing nontrivially in the Schur function expansion of P r must contain the partition (r+1, r+1), then any partition K above contains the partition (r, r). Since this last partition is not contained in the (1, r − 1)-hook, Eqs. (31) with r replaced by r − 1 are automatically fulfilled by virtue of the vanishing property (22). Note that Eq. (34) is a particular case of Eq. (32). Indeed, specializing b r−1 to x 1 +x 2 in Eq. (32), we get Eq. (34). Therefore it suffices to show that where E = 2x 1 + 2x 2 . We apply to each summand of Q(X 2 − E − B r−2 ) the factorization property (25), and divide it by the resultant Suppose that the resulting factor is: where (k 1 − 1, k 2 − 1, k 3 − 1) = (c, r + a, r + b).
Performing the same division of we get R( x 1 +x 2 , B r−2 ). Thus the wanted equation Q = P r−1 is equivalent to To prove Eq. (45) we use Eqs. (32) and (41) for P r : Using again the factorization property (this time w.r.t. the larger rectangle (r + 1) 2 ) and dividing both sides of the last equation by the resultant we get the identity and taking the coefficients of (−b r−1 ) in both sides of Eq. (46), we get the wanted Eq. (45). The lemma has been proved. ✷ (For example, writing P 3 = αS 46 + βS 55 + γS 244 + δS 145 , we get that by taking the coeficients of (−b 2 ) in both sides of

Iterating Eq. (40) gives
Corollary 13 With the above notation, we have Of course, P o r is uniquely determined by its value on X 2 . The following result gives this value.

Summing up, we have
Proposition 15 For r ≥ 1, we have where the d ij 's are defined above.

Towards Thom polynomials for A i
The following function F (i) r will be basic for computing the Thom polynomials for A i (i ≥ 1). We set where the sum is over partitions J ⊂ (r i−1 ), and for i = 1 we understand F (1)

Example 17
We have In the following, we shall tacitly assume that x, x 1 , x 2 , and B r are variables (though many results remain valid without this assumption).
The following result gives the key algebraic property of F (i) r .
(ii) Moreover, we have Proof. Substituting in Eq. (57): for p ≤ i, and, respectively, we get the assertions. ✷ Theorem 20 ([27], [26]) The polynomials S r and j≤r 2 j S r−j,r+j are Thom polynomials (with parameter r) for the singularities A 1 and A 2 .
Proof. Since only A 0 has smaller codimension than A 1 , and only A 0 , A 1 are of smaller codimension than A 2 , the equations from Theorem 1 characterizing these Thom polynomials are: for A 1 , and for A 2 . Hence the claim follows from Corollary 19. ✷ Since the singularities = A 3 , whose codimension is ≤ codim(A 3 ) are: A 0 , A 1 , A 2 and, for r ≥ 2, III 2,2 (cf. [4]), Theorem 1 yields the following equations characterizing T A 3 : By Corollary 19, the first four equations are satisfied by the function F r . For r = 1, this means that is the Thom polynomial for A 3 . However, for r ≥ 2, F (3) r does not satisfy the last vanishing, imposed by III 2,2 . In the following we shall "modify" F Let us discuss now A 4 for r = 1 (its codimension is 4). Then the singularities = A 4 , whose codimension is ≤ codim(A 4 ) are: A 0 , A 1 , A 2 , A 3 , I 2,2 . The Thom polynomial is This Thom polynomial was originally computed in [9] via the desingularization method. Its alternative derivation via solving equations imposed by the above singularities was done in [23]). It may be instructive for the reader to reprove here this result using the function F 1 . In this way we show (on this relatively simple example) the method used later to more complicated singularities. A Thom polynomial is a sum of j-polynomials (cf. Introduction) associated with the bundle morphism df : T X → f * T Y . In fact, F (i) r is the j-polynomial part for the largest possible j (the corresponding rectangle is a row of length r). Then to get the correct Thom polynomial, the function F (i) r must be modified by j-polynomials related with smaller j's. We shall see this in the next section for the singularity A 3 and r ≥ 2. In the present case, this works as follows. We have F By Corollary 19, this function satisfies the following equations imposed by However, F does not satisfy the vanishing imposed by I 2,2 . Namely, we have To see this, invoke Proposition 18: Substituting to the LHS of Eq. (70) x 1 = 0, we get by this proposition and substituting Therefore x 1 x 2 (x 1 − 2x 2 )(x 2 − 2x 2 ) divides this LHS. The coefficient −10 results from specialization x 1 = x 2 = 1. This implies Eq. (70).

Remark 21
Porteous [18] (see also [12]) gives a geometric account to the function F (i) 1 . By passing with his formulas to Schur function expansions, we should restrict ourselves to hook partitions (more precisely: to their conjugates). We have the following recursive formula (loc. cit.): Our goal now is to give an expresion for the Thom polynomial for A 3 (any r) as a linear combination of Schur functions. The cases r = 1, 2 were already known in the literature (cf., e.g., [24]). In [2], the authors announced a certain expression in terms of the Chern monomial basis. Our expression is of different form (a linear combination of Schur functions), and for the moment we do not know how to pass from it to the one in [2]. (A computer check for small values of r shows the desired coincidence.) The Thom polynomial for r = 1 has been already discussed. For r = 2, the Thom polynomial is and it differs from F by 5S 33 which is the "correction term" in this case. In the following, we shall find such a correction term for any r.
We now define the function H r (•): We state

Theorem 22
The Thom polynomial for the singularity A 3 , with parameter r, is equal to F r + H r .
We outline a proof of the theorem in the appendix. We shall now present some examples. We have the following matrix [e ij ] : Consider the following matrix whose elements are two row partitions (the symbol "∅" denotes the empty partition): We use for this matrix the same "matrix coordinates" as for the previous one. Denote by I(i, j) the partition occupying the (i, j)th place in this matrix. So, e.g., I(i, 0) = (i + 1, 2i − 1) for i ≥ 2. For r ≥ 2, we set  Then we have, with the endomorphism τ defined before Lemma 12, or iterating,

Appendix
We present here an outline of the proof of Theorem 22. The first result says that the addition of H r to F It follows from the lemma that it suffices to show that i.e. we have the vanishing imposed by III 2,2 . We look at the specialization By the factorization property (25), each polynomial S c,r+1+a,r+1+b (X 2 − D − B r−2 ) (83) factorizes into: where R = R(X 2 , D + B r−2 ). We set V r (X 2 ; B r−2 ) := H r R , so that Eq. (76) gives V r (X 2 ; B r−2 ) = e r−k,j S k (−D − B r−2 ) S j,r−k−j−2 (X 2 ) , where summation is as in Eq. (76).