List of Notations

Here I list meanings of notations that may have not been explained elsewhere.

• \(\ty\): Is for type. Given a word \(w \in [n]^\ell\), \(\ty w = (m_1, m_2, ..., m_n)\) where \(m_i\) is the number of \(i\)'s in \(w\). For example \(\ty (1, 2, 2, 1, 4, 2) = (2, 3, 0, 1)\). The definition of \(\ty T\) for a tableau \(T\) is similar.
• \([n]\): For \(n \in \pint_{>0}\), \([n]\) stands for the set \(\{1, 2, ..., n\}\).
• \(i : j\): For \(i, j \in \intg\), \(i : j\) stands for the set \(\{i, i + 1, ..., j\}\), or the sequence \((i, i + 1, ..., j)\), depending on the context.
• \(k = i : j\): Means \(k\) iterates over \(i\), \(i + 1\),..., \(j\). For example \(\sum_{k = 1 : n} a_k := \sum_{k = 1}^n a_k\).
• \(x_{i : j}\): Stands for the set \(\{x_k: k = i : j\}\) or the sequence \((x_i, x_{i + 1}, ..., x_j)\), depending on the context. So are notations like \(f(i : j)\), \(y^{i : j}\) etc.
• \(\pint\): The set of natural numbers / nonnegative integer numbers \(\{0, 1, 2,...\}\), whereas
• \(\pint_{>0}\) or \(\pint^+\): Are the set of positive integer numbers.
• \(x^w\): When both \(x\) and \(w\) are tuples of objects, this means \(\prod_i x_{w_i}\). For example say \(w = (1, 2, 2, 1, 4, 2)\), and \(x = x_{1 : 7}\), then \(x^w = x_1^2 x_2^3 x_4\).
• \(LHS\), LHS, \(RHS\), RHS: left hand side and right hand side
• \(e_i\): the \(i\)th standard basis in a vector space: \(e_i = (0, 0, ..., 0, 1, 0, 0, ...)\) where the sequence is finite or infinite depending on the dimension of the vector space and the \(1\) is the \(i\)th entry and all other entries are \(0\).