Second Orthogonality Relation

In the study of group theory, the character table serves as a vital map, translating abstract symmetries into tangible numbers. The First Orthogonality Relation reveals a profound orderliness along the rows of this table, treating irreducible characters as orthogonal vectors. But what about the columns? This question opens the door to an equally fundamental principle: the Second Orthogonality Relation. This article explores this powerful concept, which completes our understanding of the character table's symmetric structure. In the following chapters, we will first unravel the core "Principles and Mechanisms" of this relation, demonstrating how it connects character values to the group's internal architecture, such as centralizers and conjugacy classes. Subsequently, we will explore its far-reaching "Applications and Interdisciplinary Connections," showcasing its utility as a practical tool in fields from quantum mechanics to combinatorics, and even in proving foundational theorems about the existence of certain groups.

In our exploration of symmetry, we've hinted that the character table of a group is a kind of Rosetta Stone, translating the abstract language of group theory into the concrete language of numbers. We've seen that the rows of this table, each representing an irreducible character, behave like orthogonal vectors—a result known as the First Orthogonality Relation. This is a remarkable property. But nature loves symmetry, and where there is a rule for rows, it is only natural to ask: what about the columns? Is there a hidden rhythm, a secret language, written down the columns of the character table? The answer is a resounding yes, and it is just as profound and beautiful as the first.

Let’s begin our journey not with a grand theorem, but with a simple observation. Consider one of the smallest non-trivial groups, the Klein four-group $G = \mathbb{Z}_2 \times \mathbb{Z}_2$. Its elements are pairs like $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. This group has four irreducible characters (which are all one-dimensional), and its character table looks like this:

	$(0,0)$	$(1,0)$	$(0,1)$	$(1,1)$
$\chi_0$	1	1	1	1
$\chi_1$	1	-1	1	-1
$\chi_2$	1	1	-1	-1
$\chi_3$	1	-1	-1	1

The rows are clearly orthogonal, as we've discussed before. But now, run your eye down one of the columns—say, the one for the element $g_1 = (1,0)$. If we simply add up the entries, we get $1 + (-1) + 1 + (-1) = 0$. A perfect cancellation! The same happens for the other two non-identity elements. This is our first clue. Is it just a coincidence for this little group?

Let's ask a slightly more sophisticated question. The first column, corresponding to the identity element $e$, is special; its entries $\chi_i(e)$ are the dimensions of the irreducible representations. What happens if we take a "dot product" between this special first column and any other column belonging to a non-identity element $g$? This would be the sum $S = \sum_{i} \chi_i(e) \chi_i(g)$.

There is a wonderfully elegant way to see that this sum must be zero. It involves a concept called the ​​regular representation​​, which you can think of as the group acting on itself. The character of this representation, let's call it $\rho$, has a magical property: its value is the order of the group, $|G|$, at the identity element, and zero everywhere else. $\rho(e)=|G|$ and $\rho(g) = 0$ for $g \neq e$. But this "master character" can also be built from the irreducible characters, like a master chord built from individual notes. The recipe is $\rho(g) = \sum_{i} \chi_i(e) \chi_i(g)$. By comparing these two facts, we arrive at a beautiful conclusion: for any element $g$ that is not the identity, our sum must be zero. The perfect cancellation we saw was no accident; it is a deep feature of the orthogonality between the identity and every other element.

This orthogonality between the first column and the others is just the opening act. The full story—the ​​Second Orthogonality Relation​​—is a statement about any two columns. It states that for any two elements $g$ and $h$ in a group $G$:

$\sum_{i} \chi_i(g) \overline{\chi_i(h)} = \begin{cases} |C_G(g)| & \text{if } g \text{ and } h \text{ are in the same conjugacy class} \\ 0 & \text{otherwise} \end{cases}$

Let's take this apart. The sum on the left is over all irreducible characters, $\chi_i$. The term $\overline{\chi_i(h)}$ is the complex conjugate of $\chi_i(h)$. This is the standard way mathematicians define an inner product for vectors with complex entries. What this formula tells us is that the columns of the character table, when viewed as vectors, are orthogonal to each other unless they belong to the same conjugacy class!

The really fascinating part is what happens when they are not orthogonal. If we pick $g$ and $h$ from the same conjugacy class (the simplest case being $h=g$), the sum is not 1, as a naïve notion of orthogonality might suggest. Instead, it bursts forth with a number that is rich with meaning:

$\sum_{i} |\chi_i(g)|^2 = |C_G(g)|$

This $|C_G(g)|$ is the order of the ​​centralizer​​ of $g$. The centralizer is a subgroup consisting of all the elements in $G$ that commute with $g$. You can think of it as the measure of how "popular" or "symmetric" $g$ is within the group. An element that commutes with many others has a large centralizer. An element in the very center of the group, $Z(G)$, commutes with everything, so its centralizer is the entire group, $G$.

This is a stunning connection. On one side of the equation, we have a sum involving character values—numbers derived from abstract matrix representations. On the other side, we have a concrete integer: the number of elements that commute with $g$. It's as if the character table is a kind of MRI, and by summing the squares of the values in a column, we can measure a specific, tangible property of the group's internal structure.

This relation is not just an aesthetic curiosity; it is a powerful computational tool. Once you have a character table, you can deduce facts about the group's structure without the tedious work of checking commutation relations element by element.

Let's look at the quaternion group $Q_8$, the group that describes the multiplication of quaternions $\pm 1, \pm i, \pm j, \pm k$. Suppose we want to know the size of the centralizer of the element $k$. We simply look at the character table for $Q_8$, find the column for the conjugacy class of $k$, and sum the squares of its entries: $1^2 + (-1)^2 + (-1)^2 + 1^2 + 0^2 = 4$. There are precisely four elements in $Q_8$ that commute with $k$. We can do the same for the group of symmetries of a square, $D_4$. The sum of the squares of character values for a $90^\circ$ rotation $r$ is $1^2 + 1^2 + (-1)^2 + (-1)^2 + 0^2 = 4$, telling us that the centralizer of $r$ has four elements.

The formula can also be written using the size of the conjugacy class itself, since $|C_G(g)| = |G| / |C(g)|$. So, $\sum_i |\chi_i(g)|^2 = |G| / |C(g)|$. For instance, in the symmetric group $S_4$ (order 24), a 4-cycle like $(1 2 3 4)$ belongs to a conjugacy class containing 6 such cycles. The sum of squares of its character values must therefore be $24/6 = 4$.

The elegance of this principle shines brightest when we consider an element $z$ from the very center of the group. As this element commutes with everything, its centralizer is the whole group, $|C_G(z)|=|G|$. Our formula then delivers a beautifully simple result:

$\sum_{i} |\chi_i(z)|^2 = |G|$

For a central element, the sum of the squares of the character values across all irreducible characters is simply the order of the group. The most "symmetric" elements command the largest possible value from this sum.

These two orthogonality relations, for rows and for columns, are the twin pillars upon which the entire theory of characters rests. They are not independent curiosities; together, they dictate the very architecture of the character table.

If we think of the character table as an $s \times r$ matrix $M$, where $s$ is the number of irreducible characters and $r$ is the number of conjugacy classes:

• The ​​First Orthogonality Relation​​ (rows) implies, through the language of linear algebra, that the rows are linearly independent. This forces the number of rows to be no more than the number of columns: $s \le r$.
• The ​​Second Orthogonality Relation​​ (columns) implies that the columns are also linearly independent. This forces the number of columns to be no more than the number of rows: $r \le s$.

There is only one way for both of these conditions to hold true: $s$ must be equal to $r$. This is an absolutely fundamental result in group theory, yet its proof feels like a magic trick. The number of irreducible representations of a group—its fundamental modes of symmetry—is exactly equal to its number of conjugacy classes. The character table is always a ​​square matrix​​. Furthermore, since its rows (and columns) are orthogonal, it is always an ​​invertible​​ matrix. There is no redundancy; every part of the table carries essential, unique information.

The power of the second orthogonality relation doesn't stop there. It uncovers subtle relationships between a group's structure and the arithmetic nature of its character values.

For any character, the value for an inverse element, $\chi(g^{-1})$, is the complex conjugate of the value for the original element, $\overline{\chi(g)}$. Now, consider a group where some element $g$ is not conjugate to its own inverse $g^{-1}$. What can we say? The second orthogonality relation for the pair $(g, g^{-1})$ tells us that $\sum_i \chi_i(g) \overline{\chi_i(g^{-1})} = 0$. Using the property of inverses, this sum is equivalent to $\sum_i \chi_i(g) \chi_i(g) = \sum_i (\chi_i(g))^2 = 0$. A sum of squares is zero! For this to happen with complex numbers, they can't all be positive real numbers. In fact, this forces at least one of the character values $\chi_i(g)$ to be a non-real complex number. A simple structural property—the non-conjugacy of $g$ and $g^{-1}$—guarantees the existence of complex numbers in the character table!

Sometimes, this interplay involves deep results from other fields, like number theory. For the group $PSL(2,7)$, one finds that whether an element $g^2$ is conjugate to $g$ depends on whether 2 is a quadratic residue modulo 7. Because it is, the second orthogonality relation allows us to instantly evaluate a complicated-looking sum, $\sum_{\chi} \chi(g^2)\overline{\chi(g)}$, and find it must be exactly 7.

Finally, it's worth noting that the numbers in this table are not just any complex numbers. They are constrained to be ​​algebraic integers​​. A fascinating consequence is that any character value that also happens to be a rational number must be a regular integer. This is why a value like $\frac{3}{2}$ can never appear in a character table.

The second orthogonality relation, therefore, is far more than a formula. It is a governing principle, a rule of harmony that ensures the character table is a perfectly balanced, self-consistent structure. It connects the abstract world of representations to the tangible structure of the group, revealing a deep and stunning unity in the mathematics of symmetry.

Now that we have acquainted ourselves with the beautiful internal machinery of the character orthogonality relations, a natural question arises: What is this all for? Is it merely an exquisite piece of mathematical clockwork, to be admired for its internal consistency and symmetry? Or does this clockwork actually tell time? Is it a master key that unlocks doors to other rooms in the grand house of science?

The answer, you will be pleased to hear, is a resounding yes. The Second Orthogonality Relation is no mere curiosity; it is a powerful and versatile tool. It allows us to take the abstract data of a character table and convert it into concrete, structural information about a group. It acts as a bridge between the abstract world of algebra and the tangible realms of physics, chemistry, and even combinatorics. In this chapter, we will embark on a journey to explore some of these remarkable applications, and in doing so, witness the surprising power and unity of this mathematical principle.

Imagine a character table as a kind of coded blueprint for a group. At first glance, it is just a grid of numbers. But with the right key, we can decode this blueprint to reveal the group's very anatomy. The Second Orthogonality Relation is that key.

One of the most direct applications is in "measuring" the internal structure of a group. For any element $g$ in a group $G$, the relation tells us something quite profound:

$\sum_{i} |\chi_i(g)|^2 = |C_G(g)|$

The sum of the squared magnitudes of all the irreducible character values for a single element is precisely the size of that element's centralizer—the set of all elements in the group that commute with it. This is a remarkable link between the "analytic" data of the characters and the "geometric" or "structural" data of the centralizer. Once we know the size of the centralizer, we can immediately find the size of the element's conjugacy class using the Orbit-Stabilizer Theorem, $|C_g| = |G| / |C_G(g)|$.

This means we can look at a single column in a character table and, with a simple calculation, determine the size of the corresponding family of "related" elements in the group. For example, by summing the squares of the character values for a 3-cycle in the alternating group $A_5$, one can instantly discover that its centralizer has exactly 3 elements. A similar calculation using the character table for the dihedral group $D_4$ (the symmetries of a square) reveals the centralizer size for a 90-degree rotation. The abstract numbers on the page suddenly tell us how many symmetries of the square leave a particular rotation "undisturbed" by conjugation. It's like having a special kind of ruler that can measure the internal dimensions of these abstract structures.

This decoding key can also be used in reverse. Suppose we are trying to construct a character table, but one of the rows—the character of an unknown irreducible representation—is missing. The Second Orthogonality Relation, applied between the first column (the identity element) and any other column, provides a linear equation that the unknown character values must satisfy. By doing this for each column, we can generate a system of equations that often allows us to solve for the missing character values, completing the blueprint piece by piece. It's a bit like a paleontologist reconstructing an entire skeleton from just a few key bones and the universal rules of anatomy. This powerful construction method can be used to derive the final character for groups like the quaternion group $Q_8$ or the dihedral group $D_4$.

The true magic begins when we realize that the symmetries of the physical world—of molecules, crystals, and fundamental particles—are described by groups. In the strange and wonderful world of quantum mechanics, a system's states (its wavefunctions and energy levels) are classified by the irreducible representations of its symmetry group. Here, the abstract algebra of group theory becomes the concrete language of physics.

Consider the point group $D_{6h}$, which describes the symmetries of a regular hexagonal prism, a shape found in nature in everything from snowflakes to the molecular structure of benzene. This group has 24 symmetry operations, one of which is the horizontal mirror plane $\sigma_h$. What constraints does this symmetry impose on the quantum states of a benzene molecule?

The Second Orthogonality Relation provides a stunningly direct answer. Since the operation $\sigma_h$ forms a conjugacy class of size 1 in this group, the relation becomes:

$\sum_{\Gamma} |\chi_{\Gamma}(\sigma_h)|^2 = \frac{|D_{6h}|}{1} = 24$

This isn't just a mathematical statement anymore. It is a physical law. It dictates that whatever the possible quantum states (the irreducible representations $\Gamma$) of the benzene molecule are, the sum of the squares of their character values for the mirror-plane symmetry operation must equal 24. The arithmetical necessity of pure mathematics imposes a rigid constraint on the fabric of physical reality. It is a profound thought that the electrons in a molecule must conspire in such a way as to perfectly satisfy this rule, born from the abstract study of symmetry.

The orthogonality relations are not just for understanding a single group's structure; they are a bridge that connects different mathematical ideas. In a wonderful display of this unity, they allow us to answer a seemingly unrelated question from the field of combinatorics: In a finite group $G$ with $k$ conjugacy classes, how many ordered pairs of elements $(g, h)$ commute with each other?

One might start by trying to count them one by one—a truly tedious task. But there is a more elegant way. The total number of commuting pairs is, by definition, the sum of the sizes of the centralizers of all elements: $\sum_{g \in G} |C_G(g)|$.

At this point, we can make a beautiful connection. We know from the Second Orthogonality Relation that $|C_G(g)| = \sum_i |\chi_i(g)|^2$. Substituting this into our sum gives:

$\text{Number of commuting pairs} = \sum_{g \in G} \left( \sum_{i=1}^k |\chi_i(g)|^2 \right)$

Now for a classic mathematician's trick: we swap the order of summation. Instead of summing over elements first, we sum over the characters first:

$\text{Number of commuting pairs} = \sum_{i=1}^k \left( \sum_{g \in G} |\chi_i(g)|^2 \right)$

Look closely at the inner sum, $\sum_{g \in G} |\chi_i(g)|^2$. This is simply the inner product of the character $\chi_i$ with itself, which, by the First Orthogonality Relation, is equal to $|G|$. We have $k$ such terms, one for each irreducible character. So the final result is:

$\text{Number of commuting pairs} = \sum_{i=1}^k |G| = k|G|$

The result is astonishingly simple: the number of commuting pairs is just the number of conjugacy classes times the order of the group. This beautiful argument, which elegantly weaves together both orthogonality relations, takes a difficult counting problem and solves it with profound simplicity, showcasing the deep harmony within the theory.

The rigid structure imposed by the orthogonality relation is so powerful that it can be used to prove that certain mathematical objects cannot exist. It is the ultimate tool for proving impossibility. A famous example is the question of whether a non-abelian simple group of order 30 can exist.

In a wonderful piece of mathematical detective work, character theory proves this to be impossible. The argument, in essence, is a proof by contradiction. If such a group existed, its set of irreducible character degrees would be uniquely determined as $\{1, 2, 2, 2, 2, 2, 3\}$. Furthermore, since the size of any conjugacy class must divide the group's order, a class of size 5 is a theoretical possibility.

But here is where our tool comes in. Another theorem in character theory states that if the degree of a character and the size of a conjugacy class have no common factors, the character's value on that class must be zero. For a hypothetical class of size 5, its elements $g$ would be coprime to the character degrees 2 and 3. Therefore, all non-trivial characters must vanish on $g$. The Second Orthogonality sum would then be:

$\sum_i |\chi_i(g)|^2 = |\chi_{\text{trivial}}(g)|^2 + 0 + 0 + \dots = 1^2 = 1$

But the relation gives a different answer for the value of this sum: $|G|/|C_g| = 30/5 = 6$. We have arrived at the stark contradiction $1 = 6$. The numerical certainty of the orthogonality relation has proven, with unimpeachable logic, that no such group can be constructed.

The relation can also illuminate the rare and beautiful exceptions in mathematics. The symmetric group $S_6$ is famous for possessing a strange "outer" automorphism that no other symmetric group has. This automorphism shuffles the group's conjugacy classes in a peculiar way, swapping the class of single transpositions with the class of products of three disjoint transpositions. The Second Orthogonality Relation is sensitive enough to detect this. The sum $\sum_{\chi} \chi(g_1) \overline{\chi(g_2)}$ is zero if $g_1$ and $g_2$ are not conjugate. If we take $g$ to be a transposition and $h$ to be a product of three transpositions, the sum is zero. But if we apply the outer automorphism $\alpha$ to $h$, making it conjugate to $g$, the sum miraculously becomes non-zero and evaluates to the size of the centralizer of $g$. The relation acts as a precision instrument, finely tuned to the subtle concept of conjugacy.

Finally, it is worth noting that a great idea in mathematics rarely lives in isolation. Its echoes and variations often appear in other, seemingly distant fields. So it is with orthogonality. In the advanced field of modular representation theory, which studies groups in a context crucial for number theory and cryptography, one works with "Brauer characters" defined over fields with a prime characteristic $p$.

Even in this more exotic setting, an analogue of the Second Orthogonality Relation survives. It plays the same fundamental role, allowing mathematicians to deduce the structure of these "modular" representations and to construct their character tables. It is as if nature has a favorite song, and we hear its melody not just in the key of classical group theory, but also in the strange and beautiful keys of modular arithmetic. The underlying harmony, the principle of orthogonality, remains.

From mapping the internal geography of abstract groups to constraining the physical laws of the quantum world, from solving combinatorial puzzles to proving deep structural theorems, the Second Orthogonality Relation reveals itself not as a mere formula, but as a fundamental principle of symmetry. Its journey through science is a testament to the remarkable and often surprising unity of mathematical thought.