From a point group to its dictionary
In the previous guides of this rung you learned to hunt for symmetry elements and the symmetry operations tied to them — rotations, mirrors, inversion — and then to gather them up and assign a molecule to a point group like C2v, D4h, or Oh. That label is a real achievement, but on its own it is just a name. The character table is what makes the name pay off: for each point group there is one fixed table, the same in every textbook, that catalogues everything the symmetry of that group lets a molecule do. Think of the point group as the language a molecule speaks, and the character table as that language's dictionary.
The whole point of a character table is reduction in the everyday sense: instead of describing how a thing moves in full geometric detail, you boil that behavior down to a short list of standard labels. Each label answers, for one symmetry-related object, the only questions symmetry can ask: when you carry out a rotation or reflection, does this object come back unchanged, flip its sign, or get scrambled together with its partners? Two molecules that look nothing alike but belong to the same point group share the very same table — symmetry does not care what the atoms are, only how the shape behaves. That is why one C2v table serves water, sulfur dioxide, and a bent transition-metal fragment alike.
Reading the table: rows, columns, and characters
Open the C2v table and orient yourself. Across the top run the symmetry operations of the group, grouped into classes: for C2v that is E (do nothing), C2 (rotate 180 degrees about the main axis), and two mirror planes, sigma-v and sigma-v-prime. Down the left side run the row labels — the irreducible representations, the fundamental symmetry types of the group. C2v has exactly four: A1, A2, B1, and B2. These labels, called Mulliken symbols, are the alphabet of symmetry, and once you meet them in a small group like C2v you will recognize their cousins everywhere.
The numbers filling the body of the table are the characters, and for these small groups every one is simply +1 or -1. A +1 in a given column means "under that operation, an object of this symmetry type is left completely unchanged"; a -1 means "it comes back looking the same but with its sign flipped." So A1 is the fully symmetric type — all +1, unchanged by everything; that is why a label of A1 always sits at the top. B1 might read +1 under E and one mirror but -1 under C2 and the other mirror. Reading a row left to right is reading a behavioral fingerprint: this is exactly how an object of this type responds to each operation in turn.
Picture the C2v table as a small grid. The four rows are A1, A2, B1, B2; the four columns are E, C2, sigma-v(xz), sigma-v-prime(yz). A1 reads +1, +1, +1, +1 — the fully symmetric type, unchanged by everything. A2 reads +1, +1, -1, -1 — symmetric under the rotation but flipping sign in both mirrors. B1 reads +1, -1, +1, -1, and B2 reads +1, -1, -1, +1; both flip sign under the C2 rotation (that is what the B tells you) and then split on the mirrors. Run your finger across any row and you have that symmetry type's complete behavioral fingerprint — four numbers, one per operation.
The rightmost columns are the table's quiet workhorse. They tell you which everyday things transform as each row: the coordinate z follows A1, the in-plane axes x and y follow B1 and B2, the rotations Rx, Ry, Rz are listed too, and in larger tables you also find quadratic functions like z-squared and x-squared-minus-y-squared. This is the bridge from abstract labels to real chemistry. Because a pz orbital points along z, it transforms as A1; a px orbital transforms as B1. So the table hands you, for free, the symmetry label of every p orbital and d orbital on the central atom — which is precisely what you need to build a molecular orbital diagram.
Decoding the Mulliken labels
Those cryptic symbols — a1, e, t2g — are not arbitrary; each letter and subscript encodes a piece of the fingerprint, so you can almost read the behavior straight off the label. The leading letter records dimension: A and B mean a single, non-degenerate object (one orbital, standing alone); E means a doubly degenerate pair that must be treated as one unit (two orbitals locked at the same energy); T means a triply degenerate set of three. The capital letter is just the value of the character under the identity operation E — 1 for A and B, 2 for E, 3 for T — which is literally how many things travel together as a package.
The decorations refine it further. A versus B splits the non-degenerate types by how they behave under the main rotation axis: A is symmetric (+1, unchanged by the principal rotation), B is antisymmetric (-1, flips sign). Subscripts 1 and 2 distinguish behavior under a secondary axis or a vertical mirror. In groups that have a center of inversion — like the octahedral Oh — every label also wears a g or u subscript: g (from German gerade, even) means the object is unchanged by inversion through the center, u (ungerade, odd) means it flips sign. That is why the famous octahedral d-orbital labels read t2g and eg, both even, while p orbitals in the same group come out odd, as t1u.
Building a reducible representation
The real power comes when you take a set of objects that are not yet sorted by symmetry — the four O-H stretching motions of a hypothetical square molecule, the three N-H bonds of ammonia, or a clutch of orbitals on outer atoms — and ask which symmetry types they break down into. The starting point is a reducible representation, usually written with the capital Greek gamma. It is one number per operation class, and you generate it by a beautifully simple counting rule: for each symmetry operation, count how many objects in your set stay put — are not moved to a different position. That count is the character of the reducible representation for that operation.
Take ammonia, NH3, in its point group C3v, and use the three N-H bonds as our set. Under E (do nothing), all three bonds stay put, so the character is 3. Under C3 (rotate 120 degrees about the axis through nitrogen), every bond swings to where a neighbor was — none stays put — so the character is 0. Under a sigma-v mirror plane, each plane contains nitrogen and one of the three hydrogens: that one N-H bond lies in the mirror and is unmoved, while the other two swap; so exactly one stays put and the character is 1. Our reducible representation reads (3, 0, 1) across E, C3, sigma-v. That short triple is the entire symmetry content of the three bonds, captured in three numbers.
There is one refinement to be honest about for orbitals and full atomic motions rather than simple bonds. When an object stays in place but flips its own direction — say a px orbital that survives a mirror with its lobes reversed, or an atom that is unmoved by a rotation while its motion vector partly turns — it contributes not +1 but its own little character (+1 if it keeps its sense, -1 if reversed, and a cosine factor for partial rotation). For a set of plain bonds or sigma orbitals pointing outward, every object that stays put contributes a clean +1, and the simple head-count above is exactly right. Keep that distinction in your back pocket; for the cases in this rung the simple count carries you through.
Reducing it with the reduction formula
A reducible representation is a mixture; the goal is to find out which irreducible representations it is built from, and in what amounts — the way you might ask which pure notes make up a chord. The reduction formula does this. For each irreducible representation, you compute the number of times it appears, call it n, as: take each operation class, multiply three things together — how many operations are in that class, the character of your reducible representation there, and the character of the irreducible representation there — add those products across all classes, then divide by h, the total number of operations in the group (the order of the group).
n(i) = (1/h) * SUM over classes [ g(c) * X_red(c) * X_i(c) ]
h = order of the group (total # of operations)
g(c) = number of operations in class c
X_red = character of the reducible rep in class c
X_i = character of irreducible rep i in class c
NH3 example, C3v: h = 6, classes E(g=1) 2C3(g=2) 3sv(g=3)
Gamma(N-H) = ( 3, 0, 1 )
A1 = ( 1, 1, 1 )
n(A1) = (1/6)[ 1*3*1 + 2*0*1 + 3*1*1 ] = (1/6)(6) = 1
n(A2) = (1/6)[ 1*3*1 + 2*0*1 + 3*1*(-1) ] = 0
n(E) = (1/6)[ 1*3*2 + 2*0*(-1) + 3*1*0 ] = (1/6)(6) = 1
Gamma(N-H) = A1 + EThe result, Gamma equals A1 plus E, says something concrete and checkable. The three N-H bonds do not behave as three independent things; symmetry insists they reorganize into one fully symmetric combination (all three stretching in and out together — that is the A1) plus a degenerate pair of combinations where the symmetry is shared two ways (the E). A fast sanity check: the dimensions must add up. A1 counts as 1 and E counts as 2, total 3 — exactly the three bonds we started with. If your reduced reps do not add back to the size of your original set, you have made an arithmetic slip; redo the count. These symmetry-correct combinations have a name you will meet next: symmetry-adapted linear combinations.
Why chemists actually do this
This is not arithmetic for its own sake — it is the engine room beneath two whole subjects you are climbing toward. In bonding, the symmetry labels tell you which orbitals are allowed to combine: a central-atom orbital can only mix with a ligand combination that carries the same irreducible representation, because orbitals of different symmetry have exactly zero net overlap. That single rule is what lets you build a correct molecular orbital diagram for anything from methane to an octahedral complex without guessing — you match labels, and the symmetry-mismatched orbitals simply stay non-bonding. The famous t2g-below-eg splitting of d orbitals in an octahedral field is itself just two symmetry labels falling out of the Oh table.
In spectroscopy the same machinery decides what you can see. If you take the 3N motions of a molecule, reduce them, and strip off the labels belonging to translation and rotation (the table flags these — recall the x, y, z and Rx, Ry, Rz entries), what is left is the symmetry of the vibrations. Selection rules then read straight off the same table: a vibration is infrared-active only if it transforms as x, y, or z, and Raman-active only if it transforms as one of the quadratic functions. So a character table predicts, before you ever touch an instrument, how many infrared bands a molecule should show — which is how symmetry alone can distinguish, say, a square-planar XeF4 from a hypothetical tetrahedral arrangement.
Keep the honest limits in view. Symmetry tells you what is allowed and what is forbidden, never how big or how strong — it can promise that two orbitals may combine, but not the energy they end up at, nor whether a permitted band is intense or faint. It is a powerful filter, not a calculator. And its conclusions are exactly as good as the point group you assigned: get the symmetry wrong, and the cleanest reduction in the world describes the wrong molecule. With that caution, you now hold the central skill of this rung — turn a shape into a point group, look up its character table, generate a reducible representation by counting what stays put, and reduce it. Bonding diagrams and spectra are now things you can derive rather than memorize.