teach me the proof of this theorem---Theorem 7.2.3 (Jacobi-Trudi Determinants). Suppose $\lambda = (\lambda_1, \dots, \lambda_l)$.
(a) $s_\lambda = \det[h_{\lambda_i - i + j}]_{1 \le i, j \le l}$.
(b) $s_\lambda = \det[e_{\lambda_i - i + j}]_{1 \le i, j \le l}$.
Before beginning the proof, we note that a good way of remembering the subscripts in these determinants is to put the parts of $\lambda$ down the diagonal and then in each row add 1 or subtract 1 as one moves right or left, respectively. So, for example,
$s_{(7,4,1)} = \det \begin{bmatrix} h_{7} & h_{8} & h_{9} \\ h_{3} & h_{4} & h_{5} \\ h_{-1} & h_{0} & h_{1} \end{bmatrix} = \det \begin{bmatrix} h_{7} & h_{8} & h_{9} \\ h_{3} & h_{4} & h_{5} \\ 0 & 1 & h_{1} \end{bmatrix}$.
Proof. (a) We will use northeast lattice paths P in the extension of the integer lattice $\mathbb{Z}^2$ obtained by adding a vertex $(i, \infty)$ for each $i \in \mathbb{Z}$. One can only reach $(i, \infty)$ by taking an infinite number of north steps along the line $x = i$. We label the east steps of $P : s_1 s_2 s_3 \dots$ by letting $L(s_i)$ be the $y$-coordinate of $s_i$ if $s_i = E$. See, for example, the path on the left in Figure 7.3 where we are assuming the path starts at the point $(1, 1)$. If $P$ only has a finite number of east steps all on or above $y = 1$, we weight it by
$wt P = \prod_{s_i} x_{L(s_i)}$
where the product is over all $s_i$ which are east steps of P. In Figure 7.3 we have $wt P = x_1 x_3^2 x_4$.
Now let $u = (i, 1)$ and $v = (i + n, \infty)$ where $n \ge 0$. Then all $P \in \mathcal{P}(u; v)$ have exactly $n$ east steps. Furthermore, as $P$ varies over all elements of $\mathcal{P}(u; v)$ we see that
**Figure Description:**
The image displays two diagrams side-by-side. Each diagram shows a grid of points and a path connecting some of these points.
**Diagram 1 (Left):**
* **Type:** Grid of points with a path.
* **Grid:** A 5x5 grid of points arranged in a square lattice.
* **Points:** Black dots representing points.
* **Path:** A path starting from the bottom-left point, labeled (1,1). The path consists of connected straight line segments. The path moves from (1,1) one step right, then two steps up, then two steps right, then two steps up.
* **Labels on Path Segments:** Numbers are placed next to some segments of the path. From bottom-left, the labels are 1 (horizontal segment), 3 (vertical segment), 3 (horizontal segment), 4 (vertical segment).
* **Starting Point Label:** The bottom-left point is labeled "(1,1)".
**Diagram 2 (Right):**
* **Type:** Grid of points with a path.
* **Grid:** A 5x5 grid of points arranged in a square lattice, identical to the grid on the left.
* **Points:** Black dots representing points.
* **Path:** A path starting from the bottom-left point, labeled (1,1). The path consists of connected straight line segments. The path moves from (1,1) one step right, then two steps up, then two steps right, then two steps up. This path follows the same sequence of moves as the path on the left.
* **Labels on Path Segments:** Numbers are placed next to some segments of the path. From bottom-left, the labels are 1 (horizontal segment), 4 (vertical segment), 5 (horizontal segment), 7 (vertical segment).
* **Starting Point Label:** The bottom-left point is labeled "(1,1)".
**Textual Information:**
Figure 7.3. A path with the h-labeling on the left and the e-labeling on the right
wt $P$ varies over all products $x_{j_1} x_{j_2} \cdots x_{j_n}$ with $1 \le j_1 \le j_2 \le \cdots \le j_n$. It follows that wt $P(u; v) = h_{\lambda_n}$. To apply Lemma 2.5.4, let the initial and final vertices be
$$ u_i = (1 - i, 1) \quad \text{and} \quad v_i = (\lambda_i - i + 1, \infty) $$
for $i \in [l]$. See Figure 7.4 for an example where $\lambda = (3, 3, 1)$. With this choice of vertices, the weighted entries of the Lindström-Gessel-Viennot matrix give (up to transposition which does not affect the determinant) those on the right in part (a) of this theorem. Indeed, if $P$ goes from $u_i$ to $v_j$, then it has
(7.8) $$ (\lambda_j - j + 1) - (1 - i) = \lambda_j + i - j $$
east steps so that the set of such paths has weight $h_{\lambda_j + i - j}$. Also note that since the weight of a step only depends upon its height, the map $\Omega$ is weight preserving. We also need to show that any path family $P$ whose associated permutation is not the identity must be intersecting. This will be left as an exercise.
Finally, to complete the proof, we merely need to show that the weight-generating function for the nonintersecting path families is $s_\lambda$. For this it suffices to give a weight-preserving bijection from such paths to SSYT$(\lambda)$. Map such a path family $(P_1, \dots, P_l)$ to the tableau $T$ whose $i$th row consists of the labels on $P_i$ read left to right. An example is in Figure 7.4. Since $P_i$ goes from $u_i$ to $v_i$ it has $\lambda_i$ east steps by (7.8), so $T$ has shape $\lambda$. Further, the definition of the map and the labeling of the paths show that the rows are weakly increasing. To show that the columns are strictly increasing, we need to check that for all $i$ and $j$, the $j$th step on $P_i$ is lower than the $j$th step on $P_{i+1}$. But this is forced by the nonintersecting condition. It is easy to describe an inverse sending a tableau back to a path family, so we leave this detail to the reader.
**Textual Information:**
Figure 7.4. Nonintersecting paths and the associated semistandard Young tableau
(b) The proof is similar to that of (a) except that we label the east steps of $P$ by $L'(s_i) = i$. See the path on the right in Figure 7.3 for an example. The reader will find it a good exercise to fill in the rest of the proof.
**Chart Description:**
* **Type:** Diagram showing a grid of points with paths and an associated table (Young tableau).
* **Main Elements:**
* **Grid of Points:** A rectangular grid of black dots, arranged in columns and rows.
* **Column Labels:** The columns from right to left are labeled $v_1$, $v_2$, $v_3$, with vertical dotted lines extending upwards from each label, indicating potentially more points above.
* **Row Labels:** The lowest row of points from left to right is labeled $u_3$, $u_2$, $u_1$.
* **Paths:** Three distinct paths are drawn on the grid using dashed lines connecting points.
* Path 1: Starts at $u_1$, moves right, up, right, up, right. Segments are labeled '1', '1', '3'. This path appears to end in the $v_1$ column.
* Path 2: Starts at $u_2$, moves up, right, up, right. Segments are labeled '2', '4', '4'. This path appears to end in the $v_2$ column.
* Path 3: Starts at $u_3$, moves up, right, up. Segments are labeled '5'. This path appears to end in the $v_3$ column.
* **Nonintersecting Property:** The paths are drawn such that they do not cross each other.
* **Association Symbol:** A horizontal double-headed arrow ($\leftrightarrow$) is placed between the grid/paths diagram and the table structure.
* **Table (Young Tableau):** A structure resembling a Young tableau is shown to the right of the grid. It is left-justified and has 3 rows.
* Row 1: Contains three cells with numbers 1, 1, 3.
* Row 2: Contains two cells with numbers 2, 4, 4.
* Row 3: Contains one cell with number 5.
* **Labels and Annotations:** Numerical labels (1, 1, 3, 2, 4, 4, 5) are placed next to segments of the paths. The endpoints $u_1, u_2, u_3$ and column indicators $v_1, v_2, v_3$ are labeled. The figure caption describes the content.
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Welcome to our exploration of the Jacobi-Trudi determinants theorem. This important result in algebraic combinatorics relates Schur functions to determinants of complete homogeneous or elementary symmetric functions. The theorem states that for a partition lambda, the Schur function s_lambda equals the determinant of a matrix whose entries are complete homogeneous symmetric functions h with specific indices. Alternatively, it can be expressed using elementary symmetric functions e. A helpful way to remember the pattern is that the partition values appear on the diagonal of the matrix, and indices increase by 1 as you move right and decrease by 1 as you move left. For example, with lambda equals (7,4,1), we get this determinant where the diagonal entries correspond to the partition values.
Let's begin the proof of the Jacobi-Trudi formula. We'll use weighted lattice paths in the integer grid extended with points at infinity. For a northeast lattice path P, we define its weight as the product of x_L of s_i for all east steps s_i, where L of s_i is the y-coordinate of that step. The key insight is that the sum of weights of all paths from point (i,1) to point (i+n,infinity) equals the complete homogeneous symmetric function h_n. This is because such paths have exactly n east steps, and their y-coordinates form all possible weakly increasing sequences of length n. In our example, the path has weight x_1 squared times x_2, corresponding to the y-coordinates of its east steps. This connection between lattice paths and symmetric functions is the foundation of our proof.
Now we apply the Lindström-Gessel-Viennot lemma, which relates determinants to non-intersecting path families. We define our initial vertices u_i as (1-i, 1) and final vertices v_j as (lambda_j minus j plus 1, infinity). For a path from u_i to v_j, the number of east steps is lambda_j plus i minus j, so the sum of weights equals h_lambda_j plus i minus j. This gives us the LGV matrix with entries M_i,j equal to h_lambda_j plus i minus j. The determinant of this matrix equals the determinant in our theorem. By the LGV lemma, this determinant equals the sum of weights of non-intersecting path families from the u_i's to the v_i's. In our visualization, we see three non-intersecting paths, each with its own color. The key insight is that these non-intersecting path families correspond exactly to Schur functions.
To complete the proof, we need to show that the sum of weights of non-intersecting path families equals the Schur function s_lambda. We establish a weight-preserving bijection between non-intersecting path families and semistandard Young tableaux of shape lambda. Given a path family (P_1 through P_l), we map it to a tableau T where the i-th row consists of the labels of east steps in path P_i, read from left to right. In our example, the first path has east steps labeled 1, 1, and 2, which become the first row of our tableau. The second path gives us 1, 2 for the second row, and the third path gives us 1 for the third row. This bijection has several important properties: the shape of the tableau is lambda because path P_i has exactly lambda_i east steps; the rows are weakly increasing because the y-coordinates of east steps along a path never decrease; and the columns are strictly increasing because the non-intersecting condition forces the j-th east step of P_i to be at a lower y-coordinate than the j-th east step of P_(i+1). Since this bijection preserves weights, we've shown that s_lambda equals the determinant in the Jacobi-Trudi formula.
Let's summarize what we've learned about the Jacobi-Trudi determinants. The theorem expresses Schur functions as determinants of matrices whose entries are complete homogeneous symmetric functions or elementary symmetric functions. Our proof for part (a) used the Lindström-Gessel-Viennot lemma, which connects determinants to non-intersecting lattice paths. We established that the determinant equals the sum of weights of non-intersecting path families, and then created a weight-preserving bijection between these path families and semistandard Young tableaux. This completed the proof that s_lambda equals the determinant of h_lambda_i minus i plus j. For part (b), a similar approach works with a different labeling scheme for the paths. These determinantal formulas are powerful tools in algebraic combinatorics, providing efficient ways to compute and study Schur functions, which have applications in representation theory, algebraic geometry, and mathematical physics.