Iconography
Representations of Ganesha show wide variations and distinct patterns changing over time. He may be portrayed standing, dancing, heroically taking action against demons, playing with his family as a boy, sitting down or on an elevated seat, or engaging in a range of contemporary situations.

Ganesha has the head of an elephant and a big belly. This statue has four arms, which is common in depictions of Ganesha. He holds his own broken tusk in his lower-right hand and holds a delicacy, which he samples with his trunk, in his lower-left hand. The motif of Ganesha turning his trunk sharply to his left to taste a sweet in his lower-left hand is a particularly archaic feature. A more primitive statue in one of the Ellora Caves with this general form has been dated to the 7th century.[38] Details of the other hands are difficult to make out on the statue shown. In the standard configuration, Ganesha typically holds an axe or a goad in one upper arm and a noose in the other upper arm.

Associations
Obstacles
Ganesha is Vighneshvara or Vighnaraja, the Lord of Obstacles, both of a material and spiritual order. He is popularly worshipped as a remover of obstacles, though traditionally he also places obstacles in the path of those who need to be checked. Paul Courtright says that "his task in the divine scheme of things, his dharma, is to place and remove obstacles. It is his particular territory, the reason for his creation."

Krishan notes that some of Ganesha's names reflect shadings of multiple roles that have evolved over time. Dhavalikar ascribes the quick ascension of Ganesha in the Hindu pantheon, and the emergence of the Ganapatyas, to this shift in emphasis from vighnakart? (obstacle-creator) to vighnahart? (obstacle-averter). However, both functions continue to be vital to his character, as Robert Brown explains, "even after the Pur??ic Ga?e?a is well-defined, in art Ga?e?a remained predominantly important for his dual role as creator and remover of obstacles, thus having both a negative and a positive aspect

Buddhi (Knowledge)

Ganesha is considered to be the Lord of letters and learning. In Sanskrit, the word buddhi is a feminine noun that is variously translated as intelligence, wisdom, or intellect. The concept of buddhi is closely associated with the personality of Ganesha, especially in the Puranic period, when many stories stress his cleverness and love of intelligence. One of Ganesha's names in the Ganesha Purana and the Ganesha Sahasranama is Buddhipriya. This name also appears in a list of 21 names at the end of the Ganesha Sahasranama that Ganesha says are especially important. The word Priya can mean "fond of", and in a marital context it can mean "lover" or "husband", so the name may mean either "Fond of Intelligence" or "Buddhi's Husband"

Frist Chakra
According to Kundalini yoga, Ganesha resides in the first chakra, called Muladhara (m?l?dh?ra). Mula means "original, main " adhara means "base, foundation". The Muladhara chakra is the principle on which the manifestation or outward expansion of primordial Divine Force rests.[95] This association is also attested to in the Ganapati Atharvashirsa. You continually dwell in the sacral plexus at the base of the spine. Thus, Ganesha has a permanent abode in every being at the Muladhara. Ganesha holds, supports and guides all other chakras, thereby "governing the forces that propel the wheel of life"

Youtube Mantra Link

Long description
Most of what is known of the Indian kīla lore has come by way of Tibetan culture. Scholars such as F. A. Bischoff, Charles Hartman and Martin Boord have shown that the Tibetan literature widely asserts that the Sanskrit for their term phurba is kīlaya (with or without the long i). However, as Boord describes it, "all dictionaries and Sanskrit works agree the word to be kīla (or kīlaka). I suppose this [discrepancy] to result from an indiscriminate use by Tibetans of the dative singular kīlaya. This form would have been familiar to them in the simple salutation namo vajrakīlaya (homage to Vajrakīla) from which it could easily be assumed by those unfamiliar with the technicalities of Sanskrit that the name of the deity is Vajrakīlaya instead of Vajrakīla. It should also be noted that the term (vajra)kīlaya is frequently found in Sanskrit texts (as well as in virtually every kīlamantra) legitimately used as the denominative verb 'to spike,' 'transfix,' 'nail down,' etc.
Mayer (1996) contests Boord's assertion, pointing out that eminent Sanskritists such as Sakya Pandita employed Vajrakīlaya Further, he argues:
it is possible, on the other hand, that the name Vajrakīlaya as favoured by the Tibetans could in fact have been the form that was actually used in the original Indic sources, and that there is no need to hypothesize a correct form Vajrakīla". "Vajrakīlaya" could have come from the second person singular active, causative imperative, of the verb Kīl. Indigenous grammar (Pāṇini Dhātupāṭha I.557) gives to Kīl the meaning of bandha, i.e. "to bind", while Monier-Williams (285) gives the meanings "to bind, fasten, stake, pin". Hence the form kīlaya could mean "you cause to bind/transfix!", or "bind/transfix!". This, taken from mantras urging "bind/transfix", or "may you cause to bind/transfix", might have come to be treated as a noun; and the noun might then have become deified; hence Kīlaya might have started out as a deified imperative, in some ways comparable to the famous example of the deified vocative in the name Hevajra, and a not unheard of phenomenon in Sanskrit tantric literature. This suggestion is supported by Alexis Sanderson, a specialist in Sanskrit tantric manuscripts whom I consulted on this problem.[3]

Fabrication and components
The fabrication of kīla is quite diverse. Having pommel, handle, and blade, kīla are often segmented into suites of triunes[4] on both the horizontal and vertical axes, though there are notable exceptions. This compositional arrangement highlights the numerological importance and spiritual energy of the integers three (3) and nine (9). Kīla may be constituted and constructed of different materials and material components, such as wood, metal, clay, bone, gems, horn or crystal. Wooden kīla are favored by shamans for healing and energetic work.[citation needed]
Like the majority of traditional Tibetan metal instruments, the kīla is often made from brass and iron (terrestrial and/or meteoric iron. 'Thokcha' (Tibetan: ཐོག་ལྕགས, Wylie: thog lcags) means "sky-iron" in Tibetan and denote tektites and meteorites which are often high in iron content. Meteoric iron was highly prized throughout the Himalaya where it was included in sophisticated polymetallic alloys such as Panchaloha for ritual implements. The pommel of the kīla often bears three faces of Vajrakīla, one joyful, one peaceful, one wrathful, but may bear the umbrella of the ashtamangala or mushroom cap, ishtadevata (like Hayagriva), snow lion, or stupa, among other possibilities. The handle is often of a vajra, weaving or knotwork design. The handle generally has a triune form as is common to the pommel and blade. The blade is usually composed of three triangular facets or faces, meeting at the tip. These represent, respectively, the blade's power to transform the negative energies known as the "three poisons" or "root poisons" (Sanskrit: mula klesha) of attachment/craving/desire, delusion/ignorance/misconception, and aversion/fear/hate.