The Flagged Cauchy Determinant

William Y. C. Chen^{1}, Christian Krattenthaler^{2} and Arthur L. B. Yang^{3}

1,3Center for Combinatorics, LPMC Nankai University, Tianjin 300071, P. R. China

         Email: ^{1}chen@nankai.edu.cn, ^{3}arthuryang@nankai.edu.cn

2Institut Girard Desargues, Université Claude Bernard Lyon-I, 21, avenue Claude Bernard, F-69622 Villeurbanne Cedex, France

                  Email: ^{2}kratt@euler.univ-lyon1.fr

Abstract. We consider a flagged form of the Cauchy determinant, for which we provide a combinatorial interpretation in terms of nonintersecting lattice paths. In combination with the standard determinant for the enumeration of nonintersecting lattice paths, we are able to give a new proof of the Cauchy identity for Schur functions. Moreover, by choosing different starting and end points for the lattice paths, we are led to a lattice path proof of an identity of Gessel which expresses a Cauchy-like sum of Schur functions in terms of the complete symmetric functions.

Keywords: Divided difference, Cauchy identity, flagged Cauchy determinant, multi-Schur function, nonintersecting lattice paths.

AMS Classification: 05E05, 05A15.

Suggested Running Title: The Flagged Cauchy Determinant

Corresponding Author: William Y. C. Chen, chen@nankai.edu.cn

1.. Introduction

Let X = {x_{1}, x_{2}, …, x_{n}} and Y = {y_{1}, y_{2}, …, y_{n}} be two sets of variables, and let s_{λ}(X) and s_{λ}(Y) be the Schur functions indexed by a partition λ in the sets of variables X and Y, respectively. (We refer the reader to [22] or [26, Ch. 7] for all notation and definitions concerning partitions and symmetric functions.) Then the classical Cauchy identity for Schur functions can be stated as follows:

Theorem 1.1. For n ≥ 1, we have

∏_{i,j=1}^{n} \frac{1}{1 − x_{i}y_{j}} = ∑_{λ} s_{λ}(X)s_{λ}(Y),  (1.1)

where the sum ranges over all partitions of length at most n.

Classical proofs of Theorem 1.1 are by means of the Robinson–Schensted–Knuth correspondence and the Cauchy-Binet formula, respectively [22, 26]. There is also a derivation based on a matrix product involving the elementary symmetric functions as given in Macdonald [22, p. 67].

The aim of this paper is to establish a connection between the Cauchy identity and the lattice path method due to Gessel and Viennot [8, 9].

The key ingredient in our lattice path construction is a flagged form of the Cauchy determinant with respect to the sets of variables. Recall that the Cauchy determinant in the variables X and Y is the determinant

|\frac{1}{1 − x_{i}y_{j}} |_{n×n} .

It is well-known that

|\frac{1}{1 − x_{i}y_{j}} |_{n×n} = Δ(X) Δ(Y) ∏_{i,j=1}^{n} \frac{1}{1 − x_{i}y_{j}} ,  (1.2)

where we have used the common notation for the Vandermonde determinant

Δ(X) := |x_{i}^{n−j}|_{n×n} = ∏_{1≤i<j≤n}(x_{i} − x_{j}).

This paper contains the following results:

(1) We define the flagged Cauchy determinant to be

F (X, Y) = |∑_{k} h_{k−n+i}(x_{i}, …, x_{n})h_{k−n+j}(y_{j}, …, y_{n})|_{n×n} ,  (1.3)

where h_{i} is the complete symmetric function of degree i. As a first result, by simple row and column operations, we prove (see Theorem 2.1):

|\frac{1}{1 − x_{i}y_{j}} |_{n×n} = Δ(X) · Δ(Y) · F (X, Y).

(2) We provide an interpretation of the flagged Cauchy determinant F (X, Y) in terms of nonintersecting lattice paths, and we describe as well a relation with semistandard tableaux. This leads us, in particular, to the Cauchy identity (1.1).

(3) Choosing different starting and end points for the lattice paths, we obtain the equality of the flagged Cauchy determinant F (X, Y) with a determinant in complete symmetric functions in the full sets of variables X and Y:

F (X, Y) = |∑_{k} h_{k−n+i}(X)h_{k−n+j}(Y)|_{n×n} .

It should be noted that in the above formula there appear the same indices as in the flagged formula (1.3). This leads us to a lattice path interpretation of the following identity of Gessel [7, Theorem 16]:

Theorem 1.2. We have

|∑_{k} h_{k−n+i}(X)h_{k−n+j}(Y)|_{n×n} = ∑_{λ} s_{λ}(X)s_{λ}(Y),  (1.4)

where the sum ranges over all partitions of length at most n.

We remark that in Gessel’s theorem one can actually replace X and Y by infinite sets of variables. As we are going to show, our lattice path proof covers this case as well, as well as a generalization involving skew Schur functions (see Theorem 3.5).

To conclude this introduction, we note that the idea of flagged Schur functions and multi-Schur functions has proved to be very efficient in the study of Schubert polynomials in connection with divided difference operators (see Lascoux [19] and Wachs [28]). Flagged determinants with respect to the sets of variables can also be used to give simple character formulas for the symplectic groups and the orthogonal groups, see papers by Chen, Li and Louck [3], Hamel and King [14], and Okada [23].

2.. The Flagged Cauchy Determinant

Let h_{k}(x_{i}, x_{i+1}, …, x_{n}) be the complete symmetric function of degree i in x_{i}, x_{i+1}, …, x_{n} (cf. [22] or [26, Ch. 7]). The main theorem of this section establishes a relation between the Cauchy determinant and the flagged form (1.3).

Theorem 2.1. We have

|\frac{1}{1 − x_{i}y_{j}} |_{n×n} = Δ(X) · Δ(Y) · F (X, Y),  (2.5)

where F (X, Y) denotes the determinant in (1.3).

Proof. First, we express the (i, j)-entry in the Cauchy determinant as

\frac{1}{1 − x_{i}y_{j}} = ∑_{k≥0} (x_{i}y_{j})^{k} = ∑_{k≥0} h_{k}(x_{i})h_{k}(y_{j}).

Next, we recall the divided difference property of the complete symmetric functions:

\frac{h_{k}(x_{i}, …, x_{j}) − h_{k}(x_{i+1}, …, x_{j+1})}{x_{i} − x_{j+1}} = h_{k−1}(x_{i}, …, x_{j+1}).

Subtracting the (i + 1)-st row from the i-th row and dividing the resulting row by (x_{i} − x_{i+1}) for i = 1, 2, …, n − 1, we get the determinant

|\begin{matrix}∑_{k} h_{k−1}(x_{1}, x_{2})h_{k}(y_{1}) & ∑_{k} h_{k−1}(x_{1}, x_{2})h_{k}(y_{2}) & ⋯ & ∑_{k} h_{k−1}(x_{1}, x_{2})h_{k}(y_{n}) \\ ∑_{k} h_{k−1}(x_{2}, x_{3})h_{k}(y_{1}) & ∑_{k} h_{k−1}(x_{2}, x_{3})h_{k}(y_{2}) & ⋯ & ∑_{k} h_{k−1}(x_{2}, x_{3})h_{k}(y_{n}) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ ∑_{k} h_{k}(x_{n})h_{k}(y_{1}) & ∑_{k} h_{k}(x_{n})h_{k}(y_{2}) & ⋯ & ∑_{k} h_{k}(x_{n})h_{k}(y_{n}) \end{matrix}| .

We continue by subtracting the (i + 1)-st row from the i-th row and dividing by (x_{i} − x_{i+2}) for i = 1, 2, …, n − 2, then subtracting the (i + 1)-st row from the i-th row and dividing by (x_{i} − x_{i+3}) for i = 1, 2, …, n − 3, etc. Eventually, we obtain the determinant

|∑_{k} h_{k−n+i}(x_{i}, …, x_{n})h_{k}(y_{j})|_{n×n} .

Now we apply analogous operations to the columns of the above determinant. That is, we subtract the (i + 1)-st column from the i-th column and divide the resulting column by (y_{i} − y_{i+1}) for i = 1, 2, …, n − 1, then we subtract the (i + 1)-st column from the i-th column and divide by (y_{i} − y_{i+2}) for i = 1, 2, …, n − 2, and so on. We finally get the following flagged determinant of complete symmetric functions in X and Y:

|∑_{k} h_{k−n+i}(x_{i}, …, x_{n})h_{k−n+j}(y_{j}, …, y_{n})|_{n×n} .  (2.6)

This is exactly the determinant F (X, Y). On the other hand, the division operations yield the product of the Vandermonde determinants Δ(X) and Δ(Y). This completes the proof.

3.. Lattice paths and the Cauchy identity

The lattice path method introduced by Gessel and Viennot [8, 9] (but actually dating back to Karlin and McGregor [15, 16] and Lindström [20]) has been widely used as a powerful technique for the study of symmetric functions, plane partitions and many other combinatorial problems (see for example [1, 2, 4, 6, 10, 11, 12, 13, 18, 23, 27, 29]).

We let the underlying (lattice) digraph D be the integer lattice ℤ × ℤ, where the arcs (or steps) are horizontal, (i, j) → (i + 1, j), or vertical, (i, j) → (i, j ± 1), with the following restrictions: if a vertical arc lies strictly to the left of the y-axis, it must be an up step from (i, j) to (i, j + 1); if a vertical arc lies strictly to the right of the y-axis, then it must be a down step from (i, j) to (i, j − 1); and there are no vertical steps on the y-axis. From now on, when we speak of a (lattice) path then we always mean a path in D.

We define the following weights for the arcs in D:

(1) A horizontal arc has weight 1.

(2) For i < 0, a vertical arc from (i, j) to (i, j + 1) has weight x_{n+i+1}.

(3) For i > 0, a vertical arc from (i, j) to (i, j − 1) has weight y_{n−i+1}.

The weight of a path P, denoted by w(P), is defined as the product of the weights of the arcs of the path P. Given an n-tuple (P_{1}, P_{2}, …, P_{n}) of lattice paths, its weight is defined to be the product of the weights of the path P_{i}. We now suppose that A_{1}, A_{2}, …, A_{n} and B_{1}, B_{2}, …, B_{n} are given points in the integer lattice ℤ × ℤ. Let P(A_{i}, B_{j}) denote the set of lattice paths from A_{i} to B_{j} in D. Similarly, we use P(A, B) to denote the set of all n-tuples (P_{1}, P_{2}, …, P_{n}) of lattice paths in D where P_{i} starts at A_{i} and ends at B_{i}. We also adopt the notation P_{0}(A, B) for the set of all n-tuples (P_{1}, P_{2}, …, P_{n}) of nonintersecting lattice paths, where P_{i} has starting point A_{i} and end point B_{i}. Here, as usual, by “nonintersecting” we mean that there are no common points among P_{i} and P_{j} for i ≠ j. By GF(P(A, B)) and GF(P_{0}(A, B)) we denote the generating functions, or the sums of weights, of the n-tuples of lattice paths in P(A, B) and P_{0}(A, B), respectively.

For the purpose of the promised combinatorial interpretation of the flagged Cauchy determinant F (X, Y), we choose

A_{i} = (i − n − 1, −i), and B_{i} = (n − i + 1, −i), i = 1, 2, …, n.  (3.7)

We refer the reader to Figure 3.1 for an illustration of these points in the case n = 4. (The paths should be ignored at this point.)

With the above choice, we arrive at the following lattice path interpretation of the entries in the flagged Cauchy determinant.

Lemma 3.1. The generating function for the D-paths from A_{i} to B_{j} equals

GF(P(A_{i}, B_{j})) = ∑_{k} h_{k−n+i}(x_{i}, …, x_{n})h_{k−n+j}(y_{j}, …, y_{n}).  (3.8)

Proof. We classify the D-paths P from A_{i} to B_{j} by their intersection points with the y-axis. To be more specific, assume that P intersects the y-axis in the point Q. Let k be the y-coordinate of Q. Since there are no arcs on the y-axis, the weights of all such paths sum to

h_{k+i}(x_{i}, …, x_{n})h_{k+j}(y_{j}, …, y_{n}).

Summing over k, one gets the right-hand side of (3.8), after having done the replacement k → k − n. This completes the proof of the lemma.

As an immediate corollary of the standard theorem on nonintersecting lattice paths (see [9, Cor. 2] or [27, Theorem 1.2]), we obtain the promised interpretation of the flagged Cauchy determinant in terms of nonintersecting lattice paths. We refer the reader to Figure 3.1 for an example of a set of nonintersecting lattice paths as in the statement of the theorem below (i.e., with the starting and points as in (3.7)), in the case n = 4.

Theorem 3.2. We have the following relation:

F (X, Y) = GF(P_{0}(A, B)).  (3.9)

Proof. The aforementioned theorem on nonintersecting lattice paths says that

GF(P_{0}(A, B)) = det(GF(P(A_{i}, B_{j})))_{n×n},

under the condition that the starting and end points are “D-compatible” (in the terminology of [27]), i.e., for i < j and k < l any path from A_{i} to B_{l} and any path from A_{j} to B_{k} have a point in common. However, the latter is entirely obvious from the topology of the digraph D. Hence, in view of Lemma 3.1, the theorem follows.

Given the sets A and B of starting and end points, respectively, we may translate an n-tuple (P_{1}, P_{2}, …, P_{n}) of nonintersecting lattice paths into a pair of semistandard tableaux of the same shape with entries from {1, 2, …, n}.

[figure: x; y; A_{1}; A_{2}; A_{3}; A_{4}; B_{1}; B_{2}; B_{3}; B_{4}; O; x_{1}; x_{1}; x_{2}; x_{3}; x_{3}; x_{3}; x_{4}; y_{1}; y_{2}; y_{2}; y_{3}; y_{3}; y_{4}; y_{4}]

Figure 3.1 Nonintersecting paths from A_{i} to B_{i}

S = 1 1 2 3 3 3 4 , T = 1 2 3 4 2 3 4

Figure 3.2 A pair of tableaux of the same shape

Theorem 3.3. There is a one-to-one correspondence between n-tuples (P_{1}, P_{2}, …, P_{n}) of nonintersecting lattice paths, where P_{i} runs from A_{i} to B_{i}, and pairs of semistandard tableaux of the same shape with entries from {1, 2, …, n}. In particular, we have

GF(P_{0}(A, B)) = ∑_{λ,ℓ(λ)≤n} s_{λ}(X)s_{λ}(Y).  (3.10)

Proof. While reading this proof, it is advisable to consult in parallel the example in Figures 3.1 and 3.2. (There, we have chosen n = 4.) Given any n-tuple (P_{1}, P_{2}, …, P_{n}) of nonintersecting paths such that P_{i} runs from A_{i} to B_{i}, let Q_{i} be the intersection point of P_{i} with the y-axis. We now cut each P_{i} into two segments U_{i} and V_{i}, where U_{i} runs from A_{i} to Q_{i} and V_{i} runs from Q_{i} to B_{i}. To the ntuple (U_{1}, U_{2}, …, U_{n}) we can associate a semistandard tableau S with entries from {1, 2, …, n}. This is done as follows: the i-th row of S is obtained from the path U_{i} by reading the indices of the weights of the vertical steps. (That is, we read i for a vertical step with weight x_{i}. Compare Figures 3.1 and 3.2.) The column-strictness of S is guaranteed by the fact that the paths U_{1}, U_{2}, …, U_{n} are nonintersecting. Similarly, the n-tuple (V_{1}, V_{2}, …, V_{n}) corresponds to a semistandard tableau T with entries from {1, 2, …, n}. Thus, the n-tuple (P_{1}, P_{2}, …, P_{n}) of nonintersecting lattice paths corresponds to a pair of tableaux (S, T) of the same shape. Clearly, the above procedure is reversible. Hence we obtain the claimed bijection.

[figure: O; x; y; A′_{1}; A′_{2}; A′_{3}; A′_{4}; B′_{4}; B′_{1}; B′_{2}; B′_{3}; x_{1}; x_{1}; x_{2}; x_{3}; x_{3}; x_{3}; x_{4}; y_{1}; y_{2}; y_{2}; y_{3}; y_{3}; y_{4}; y_{4}]

Figure 3.3 Nonintersecting paths from A′_{i} to B′_{i}

The identity (3.10) follows now by using the fact that the generating function ∑_{S} w(S), where the sum is over all semistandard tableaux S of shape λ with entries from {1, 2, …, n}, and where

w(S) = ∏_{i=1}^{n} x_{i}^{#(i’s in S)},

is the Schur function s_{λ}(X).

A combination of Theorems 2.1, 3.2, and 3.3, and the evaluation (1.2) of the Cauchy determinant yields Cauchy’s identity (Theorem 1.1).

We remark that, in fact, the flagged Cauchy determinant can be written as a determinant of complete symmetric functions in the full sets of variables X and Y. Let A′_{i} = (−n, −i) and B′_{i} = (n, −i). (See Figure 3.3 for the location of these points in the case that n = 4. At this point, the paths should be ignored.) It is easy to see that there is a one-to-one correspondence between n-tuples (P_{1}, P_{2}, …, P_{n}) of nonintersecting lattice paths with P_{i} running from A_{i} to B_{i} and n-tuples (P′_{1}, P′_{2}, …, P′_{n}) of nonintersecting lattice paths with P′_{i} running from A′_{i} to B′_{i}, for, restricted by the property that paths must be nonintersecting, the path P′_{i} must pass the points A_{i} and B_{i}, i = 1, 2, …, n; moreover, there is a unique way to extend the path P_{i} to the points A′_{i} and B′_{i}. Figure 3.3 shows the set (P′_{1}, P′_{2}, …, P′_{n}) of modified paths for our example in Figure 3.1. Thus, we have the identity

GF(P_{0}(A, B)) = GF(P_{0}(A′, B′)).  (3.11)

On the other hand, for the generating function of paths from A′_{i} to B′_{j} we have the following result.

Lemma 3.4. Let A′_{i} = (−n, −i) and B′_{i} = (n, −i). The generating function for paths from A′_{i} to B′_{j} equals

GF(P(A′_{i}, B′_{j})) = ∑_{k} h_{k−n+i}(X)h_{k−n+j}(Y).  (3.12)

Thus, using the standard theorem on nonintersecting paths (see [9, Cor. 2] or [27, Theorem 1.2]) another time, we obtain

GF(P_{0}(A′, B′)) = |∑_{k} h_{k−n+i}(X)h_{k−n+j}(Y)|_{n×n} .  (3.13)

A combination of Theorem 3.3 and Eqs. (3.11) and (3.13) then yields Theorem 1.2 of Gessel. We remark that our lattice path interpretation can also prove the general form of Gessel’s theorem, i.e., Theorem 1.2 where X and Y are infinite sets of variables. To accomplish the proof, one just has to “move” the starting points “to the left” and the end points “to the right,” more precisely, we would choose A′_{i} = (−∞, −i) and B′_{i} = (+∞, −i), i = 1, 2, …, n, and modify the labelling and the weights of the vertical steps accordingly.

More generally, let α and β be two partitions of length at most n. If we choose as starting points the points A′′_{i} = (−∞, α_{i} − i) and as end points the points B′′_{i} = (+∞, β_{i} − i), then, in the same way as above, we obtain the following generalization of Theorem 1.2, also due to Gessel [7, Theorem 16; cf. the paragraph just before Theorem 16].

Theorem 3.5. We have

|∑_{k} h_{k−α_{i}+i}(X)h_{k−β_{j}+j}(Y)|_{n×n} = ∑_{λ} s_{λ/α}(X)s_{λ/β}(Y),  (3.14)

where the sum ranges over all partitions of length at most n, and where s_{λ/α}(X) denotes the skew Schur function of shape λ/α in the variables X (see [22] or [26, Ch. 7]), and similarly for s_{λ/β}(Y).

Gessel’s proof of this theorem (as well as the proof of Theorem 1.2) is algebraic, as it makes use of the Cauchy–Binet theorem. For a different bijective proof, which makes use of the combinatorics of two-rowed arrays and the skew Robinson– Schensted–Knuth correspondence due to Sagan and Stanley [25], in the formulation of Fomin [5] and Roby [24, Section 4.1], see [17, proof of Theorem 3].

As we observed earlier, Theorem 1.2 and Theorem 3.2 present two different determinants for F (X, Y), which implies that these two determinants are equivalent. To conclude, we present an algebraic proof of this fact by using knowledge about multi-Schur functions. In particular, we need the following property of these functions [19, 21].

Lemma 3.6. For any families L_{0}, L_{1}, …, L_{n−1} of variables such that |L_{i}| ≤ i, we have

s_{λ}(H_{1}, H_{2}, …, H_{n}) = |h_{λ_{j}+j−i}(H_{j})|_{n×n} = |h_{λ_{j}+j−i}(H_{j} − L_{n−i})|_{n×n},  (3.15)

where H_{1}, H_{2}, …, H_{n} are sets of variables, and the complete super symmetric function h_{k}(X − Y) is defined by the generating function

∑_{k≥0} h_{k}(X − Y) t^{k} = \frac{∏_{y∈Y} (1 − yt)}{∏_{x∈X}(1 − xt)}.

We note that the matrix in Equation (1.4) can be expressed as the product of two matrices:

(∑_{k} h_{k−n+i}(X)h_{k−n+j}(Y))_{n×n} = (h_{j−i}(X))_{n×∞} · (h_{i−j}(Y))_{∞×n} .  (3.16)

Let X_{i} = {x_{1}, x_{2}, …, x_{i}} and Y_{i} = {y_{1}, y_{2}, …, y_{i}}. On the left-hand side of (3.16) we may replace the pair of sets of variables (X, Y) in the (i, j)-entry by (X − X_{n−i}, Y − Y_{n−j}). In accordance with this substitution on the left-hand side, we must make the corresponding substitutions on the right-hand side of (3.16), that is, we must replace X in the i-th row by X − X_{n−i} in the first matrix and we must replace Y in the j-th column by Y − Y_{n−j} in the second matrix. After these substitutions, Equation (3.16) becomes

(∑_{k} h_{k−n+i}(X − X_{n−i})h_{k−n+j}(Y − Y_{n−j}))_{n×n}

= (h_{j−i}(X − X_{n−i}))_{n×∞} · (h_{i−j}(Y − Y_{n−j}))_{∞×n} .  (3.17)

We now apply the Cauchy-Binet formula to (3.16), to obtain the identity

|∑_{k} h_{k−n+i}(X)h_{k−n+j}(Y)|_{n×n}

= ∑_{1≤k_{1}<···<k_{n}} |h_{k_{j}−1+j−i}(X)| · |h_{k_{i}−1+i−j}(Y)| .  (3.18)

On the other hand, the Cauchy-Binet formula applied to (3.17) yields the identity

|∑_{k} h_{k−n+i}(X − X_{n−i})h_{k−n+j}(Y − Y_{n−j})|_{n×n}

= ∑_{1≤k_{1}<···<k_{n}} |h_{k_{j}−1+j−i}(X − X_{n−i})| · |h_{k_{i}−1+i−j}(Y − Y_{n−j})| .

From Lemma 3.6 it follows that

|h_{k_{j}−1+j−i}(X)|_{n×n} = |h_{k_{j}−1+j−i}(X − X_{n−i})|_{n×n} ,  (3.19)
|h_{k_{i}−1+i−j}(Y)|_{n×n} = |h_{k_{i}−1+i−j}(Y − Y_{n−j})|_{n×n} .  (3.20)

Applying (3.19) and (3.20) to (3.18), we have

|∑_{k} h_{k−n+i}(X)h_{k−n+j}(Y)|_{n×n}

= ∑_{k_{1}<···<k_{n}} |h_{k_{j}−1+j−i}(X − X_{n−i})| · |h_{k_{i}−1+i−j}(Y − Y_{n−j})|

= |∑_{k} h_{k−n+i}(X − X_{n−i})h_{k−n+j}(Y − Y_{n−j})|_{n×n}

= |∑_{k} h_{k−n+i}(x_{i}, …, x_{n})h_{k−n+j}(y_{j}, …, y_{n})|_{n×n} .

The last equality comes from simultaneously reversing the order of rows and columns of the determinant. Therefore, we have accomplished an algebraic proof of the equality of the flagged Cauchy determinant (3.9) and the determinant (1.4) in the full sets of variables.

In the same way, we may derive a more general theorem:

Theorem 3.7. For any two families L_{0}, L_{1}, …, L_{n−1} and G_{0}, G_{1}, …, G_{n−1} of variables such that |L_{i}| ≤ i, |G_{i}| ≤ i, we have

|∑_{k} h_{k−n+i}(X)h_{k−n+j}(Y)|_{n×n}

= |∑_{k} h_{k−n+i}(X − L_{i−1})h_{k−n+j}(Y − G_{j−1})|_{n×n} .  (3.21)

Acknowledgments. This work was done under the auspices of the 973 Project on Mathematical Mechanization of the Ministry of Science and Technology, and the National Science Foundation of China. We thank Professor Alain Lascoux for his lectures on symmetric functions.

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