Скачать или смотреть Hidden Mathematics in Group Theory | Vector Space

For related videos, refer to the following links;

For the structure of the character table,    • Structure of the Character Table | Group T...  

For the physical significance of irreducible representation,    • Physical Significance of Irreducible Repre...  

For the significance of Mulliken symbols PART 1,    • Significance of Mulliken Symbols PART 1 | ...  

For the significance of Mulliken symbols PART 2,    • Significance of Mulliken Symbols Part 2 | ...  

For the mixed and closed functions in group theory,    • Mixed and Closed Functions in Group Theory...  

For the reducible representation Part 1,    • Reducible Representation | Physical Signif...  

In this lecture, we go deeper into the mathematical structure behind reducible and irreducible representations in molecular symmetry.

We often say that “representations behave like vectors in symmetry space.”
But what does that actually mean?

To understand this properly, we build the concept using a three-layer picture:

Layer 1 – Physical Function Space
Orbitals like px, py, and pz are real vectors in real physical space. These are the objects that actually move and mix under symmetry operations.

Layer 2 – Matrix Representation
Each symmetry operation is assigned a matrix that acts on those functions. The size of the matrix depends on how many basis functions we have. One function gives a 1×1 matrix, two functions give a 2×2 matrix, and so on.

Layer 3 – Character Space
Instead of working with full matrices, we compress each matrix to its trace — the character.
Now each representation becomes a list of numbers (one per class), and this list behaves like a vector.

When we say “representations form a vector space,” we are mainly referring to this third layer: the space of characters (class-function space).

In this video, we clarify:

• Why representations can be treated like vectors
• Why characters form a vector space
• Why the number of components equals the number of classes
• The difference between matrix dimension and vector-space dimension
• Why irreducible representations act as basis vectors
• How reducible representations are larger vector spaces built from irreducible ones
• What it really means to write τ = τ₁ + τ₂ + τ₃

We also use C₂v as a concrete example to remove confusion between:

Dimension of the representation (matrix size)

Dimension of character space (number of classes)

By the end of this lecture, you will clearly understand how:

Irreducible representations are basis vectors

Reducible representations are direct sums of irreducible ones

Character tables are structured as vector spaces

In the next video, we will go further and discuss:

• The physical meaning of addition (τ = τ₁ + τ₂ + τ₃)
• The meaning of multiplication (direct product)
• And how all this connects to the transition moment integral

If you want a deeper and more structural understanding of group theory in chemistry, this series is for you.