Q-Analogue And Combinatorial Proof Of Identities On The Vector Space

Let V_n(q) be a n-dimensional vector space over the finite field GF(q) with q elements, (?)n(q) its lattice of subspaces, and Bn the Boolean lattice of subsets of an n-element set {1,2, … , n}. The q-analogue between (?)n(q) and Bn means that some qualities and identities on the Boolean lattice are extended onto the subspace-lattice, then their q-analogue are discovered on the subspace-lattice, where q is an parameter. While taking the limit q → 1, the q-analogues become corresponding qualities on the Boolean lattice.Moreover, by the combinatorial proof or combinatorial interpretation, the identity is equipped with certain count meaning. The most general way in the combinatorial proofs is to count two sides of the identity by two different methods. Generaly, through building a bijection from one set to another one, the number of the two sets respectively represents the two sides of the identity. Because of the 1-1 property of bijection, the identity is proved. This thesis just applies this method to give identities their combinatorial proof. As a result, a general approach to combinatorial proof is illustrated.In the thesis some vital identities with binomial coefficient are given with their combinatorial proof, and the q-analogues of some identities are offered with corresponding combinatorial proofs on the subspace-lattice. Especially, one remarkable result is that a new q-analogue of the cube-sums identity is obtained with its combinatorial proof over the vector space.The main content of this thesis can be summarized as follows:1. Introduce relevant basic knowledge and concept about the q-analogue on the vector space, such as Poset, Lattice, Combinatorial Proof, Binomial coefficient and so on.2. Some important identitis are provided with their combinatorial proof or combinatorial interpretation on the Boole Lattice B_n.3. Introduce the concept of q-analogue. q-analogues of some important identities are given with their combinatorial proofs on the vector space. We also offer an important method of Subset-Subspace analogy and introduce the multiset Mahonian statistics.4. On the vector space a new q-analogue of the identity obtained with its Combinatorial proof.